Chapter 18 : Maxima and Minima - Rd Sharma Solutions for Class 12-science Maths CBSE

Chapter 18 - Maxima and Minima Excercise Ex. 18.1

Question 1

Solution 1

Question 2

Solution 2

Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6

Solution 6

Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9

Chapter 18 - Maxima and Minima Excercise Ex. 18.2

Question 1
Solution 1
Question 2
Solution 2

Question 3
Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6
Solution 6
Question 7

Solution 7

Question 8

Solution 8

Question 9
Solution 9
Question 10
Solution 10

Question 11

Solution 11

f open parentheses x close parentheses equals 2 sin x space minus space x comma space minus straight pi over 2 space less or equal than x space less or equal than straight pi over 2

F o r space c h e c k i n g space t h e space m i n i m a space a n d space m a x i m a comma space w e space h a v e
f apostrophe open parentheses x close parentheses equals 2 cos x space minus 1 space equals space 0

rightwards double arrow cos x equals 1 half equals cos straight pi over 3
rightwards double arrow x equals minus straight pi over 3 comma space straight pi over 3
A t space x equals minus straight pi over 3 comma space f open parentheses x close parentheses space c h a n g e s space f r o m space minus v e space t o space plus space v e
rightwards double arrow x equals minus straight pi over 3 space i s space p o i n t space o f space l o c a l space m i n i m a space w i t h space v a l u e space equals space minus square root of 3 minus straight pi over 3

A t space x equals straight pi over 3 comma space f open parentheses x close parentheses space c h a n g e s space f r o m space plus v e space t o space plus space v e
rightwards double arrow x equals straight pi over 3 space i s space p o i n t space o f space l o c a l space m a x i m a space w i t h space v a l u e space equals space square root of 3 minus straight pi over 3

Question 12
Solution 12
Question 13

Solution 13

Question 14

Solution 14

Chapter 18 - Maxima and Minima Excercise Ex. 18.3

Question 1
Solution 1
Question 2
Solution 2
Question 3

Solution 3

Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8

Solution 8

L o c a l space M a x i m u m space v a l u e space equals space f left parenthesis 4 right parenthesis
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 4 square root of 32 minus 4 squared end root
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 4 square root of 32 minus 16 end root
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 4 square root of 16
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 16
L o c a l space m i n i m u m space a t space x equals minus 4 ;
L o c a l space M i n i m u m space v a l u e space equals space f left parenthesis minus 4 right parenthesis
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals minus 4 square root of 32 minus open parentheses minus 4 close parentheses squared end root
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals minus 4 square root of 32 minus 16 end root
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals minus 4 square root of 16
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals minus 16

Question 9

Solution 9

Question 10
Solution 10
Question 11
Solution 11
Question 12
Solution 12
Question 13
Solution 13
Question 14
Solution 14
Question 15
Solution 15
Question 16

Solution 16

Question 17

Solution 17

Question 18
Solution 18
Question 19
Solution 19
Question 20

If f(x) = x3 + ax2 + bx + c has a maximum at x = -1 and minimum at x = 3. Determine a, b and c.

Solution 20

Chapter 18 - Maxima and Minima Excercise Ex. 18.4

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8

Chapter 18 - Maxima and Minima Excercise Ex. 18.5

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4

Divide 15 into two parts such that the square of one multiplied with the cube of the other is maximum.

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7
Solution 7
Question 8

Solution 8

W e space h a v e comma space 4 l plus 3 a equals 20
rightwards double arrow 4 l equals 20 minus 3 a
rightwards double arrow l equals fraction numerator 20 minus 3 a over denominator 4 end fraction
F r o m space left parenthesis i right parenthesis comma space w e space h a v e comma
s equals open parentheses fraction numerator 20 minus 3 a over denominator 4 end fraction close parentheses squared plus fraction numerator square root of 3 over denominator 4 end fraction a squared
fraction numerator d s over denominator d a end fraction equals 2 open parentheses fraction numerator 20 minus 3 a over denominator 4 end fraction close parentheses open parentheses fraction numerator minus 3 over denominator 4 end fraction close parentheses plus 2 a cross times fraction numerator square root of 3 over denominator 4 end fraction
T o space f i n d space t h e space m a x i m u m space o r space m i n i m u m comma space fraction numerator d s over denominator d a end fraction equals 0
rightwards double arrow 2 open parentheses fraction numerator 20 minus 3 a over denominator 4 end fraction close parentheses open parentheses fraction numerator minus 3 over denominator 4 end fraction close parentheses plus 2 a cross times fraction numerator square root of 3 over denominator 4 end fraction equals 0
rightwards double arrow minus 3 open parentheses 20 minus 3 a close parentheses plus 4 a square root of 3 equals 0
rightwards double arrow minus 60 plus 9 a plus 4 a square root of 3 equals 0
rightwards double arrow 9 a plus 4 a square root of 3 equals 60
rightwards double arrow a open parentheses 9 plus 4 square root of 3 close parentheses equals 60
rightwards double arrow a equals fraction numerator 60 over denominator 9 plus 4 square root of 3 end fraction
D i f f e r e n t i a t i n g space o n c e space a g a i n comma space w e space h a v e comma
fraction numerator d squared s over denominator d a squared end fraction equals fraction numerator 9 plus 4 square root of 3 over denominator 8 end fraction greater than 0
T h u s comma space t h e space s u m space o f space t h e space a r e a s space o f space t h e space s q u a r e space a n d space t r i a n g l e space i s space m i n i m u m space w h e n space a equals fraction numerator 60 over denominator 9 plus 4 square root of 3 end fraction
W e space k n o w space t h a t comma space l equals fraction numerator 20 minus 3 a over denominator 4 end fraction
rightwards double arrow l equals fraction numerator 20 minus 3 open parentheses fraction numerator 60 over denominator 9 plus 4 square root of 3 end fraction close parentheses over denominator 4 end fraction
rightwards double arrow l equals fraction numerator 180 plus 80 square root of 3 minus 180 over denominator 4 open parentheses 9 plus 4 square root of 3 close parentheses end fraction
rightwards double arrow l equals fraction numerator 20 square root of 3 over denominator 9 plus 4 square root of 3 end fraction

Question 9
Solution 9
Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

M a x i m u m space v o l u m e space i s space V subscript x equals 3 end subscript equals 3 cross times open parentheses 18 minus 2 cross times 3 close parentheses squared
rightwards double arrow V equals 3 cross times 12 squared
rightwards double arrow V equals 3 cross times 144
rightwards double arrow V equals 432 space c m cubed
Question 13

Solution 13

Question 14
Solution 14
Question 15

Solution 15

Question 16
Solution 16

Question 17
Solution 17
Question 18
Solution 18
Question 19
Solution 19
Question 20
Solution 20
Question 21

Solution 21

 

 

Question 22

An isosceles triangle of vertical angle 2θ is inscribed in a circle radius a. show that the area of the triangle is maximum when

Solution 22

 

 

Question 23
Solution 23
Question 24

Solution 24

Question 25
Solution 25
Question 26

Solution 26

Question 27

Solution 27

S i n c e space fraction numerator d squared S over denominator d x squared end fraction less than 0 comma space t h e space s u m space i s space l a r g e s t space w h e n space x equals y equals fraction numerator r over denominator square root of 2 end fraction

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

A given quantity of metal is to be cast into a half cylinder with a rectangular base and semicircular ends. Show that in order that the total surface area may be minimum, the ratio of the length of the cylinder to the diameter of its semi - circular ends is π : ( π + 2) 

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 45

Solution 45

Chapter 18 - Maxima and Minima Excercise MCQ

Question 1

The maximum value of x1/x, x > 0 is

begin mathsize 12px style left parenthesis straight a right parenthesis space straight e to the power of 1 divided by straight e end exponent
left parenthesis straight b right parenthesis space open parentheses 1 over straight e close parentheses to the power of straight e
left parenthesis straight c right parenthesis space 1
left parenthesis straight d right parenthesis space none space of space these end style

Solution 1

begin mathsize 12px style Correct space option colon thin space left parenthesis straight a right parenthesis
straight f open parentheses straight x close parentheses equals straight x to the power of 1 over straight x end exponent
taking space log space on space both space sides
logf open parentheses straight x close parentheses equals logx to the power of 1 over straight x end exponent
logf open parentheses straight x close parentheses equals 1 over straight x logx
fraction numerator 1 over denominator straight f left parenthesis straight x right parenthesis end fraction straight f apostrophe left parenthesis straight x right parenthesis equals 1 over straight x cross times 1 over straight x plus logx open parentheses fraction numerator negative 1 over denominator straight x squared end fraction close parentheses
straight f apostrophe left parenthesis straight x right parenthesis equals straight f left parenthesis straight x right parenthesis open parentheses 1 over straight x squared minus logx over straight x squared close parentheses
Consider comma
straight f apostrophe left parenthesis straight x right parenthesis equals 0
straight f left parenthesis straight x right parenthesis open parentheses 1 over straight x squared minus logx over straight x squared close parentheses equals 0
1 minus logx equals 0
straight x equals straight e
straight f apostrophe left parenthesis straight x right parenthesis equals straight f left parenthesis straight x right parenthesis open parentheses 1 over straight x squared minus logx over straight x squared close parentheses
straight f apostrophe left parenthesis straight x right parenthesis equals straight f left parenthesis straight x right parenthesis open parentheses fraction numerator 1 minus logx over denominator straight x squared end fraction close parentheses equals straight x to the power of 1 over straight x minus 2 end exponent open parentheses 1 minus logx close parentheses
straight f apostrophe apostrophe open parentheses straight x close parentheses equals straight x to the power of 1 over straight x end exponent open parentheses 1 over straight x squared minus 1 over straight x squared logx close parentheses squared plus straight x to the power of 1 over straight x end exponent open parentheses fraction numerator negative 2 over denominator straight x cubed end fraction plus 2 over straight x cubed logx minus 1 over straight x cubed close parentheses
straight f apostrophe apostrophe open parentheses straight e close parentheses equals negative straight e to the power of 1 over straight e end exponent cross times straight e to the power of negative 3 end exponent less than 0
Maximum space value space straight f open parentheses straight e close parentheses equals straight e to the power of 1 over straight e end exponent end style

Question 2

begin mathsize 12px style If space ax plus straight b over straight x greater or equal than straight c space for space all space positive space straight x space where space straight a comma straight b comma greater than 0 comma space then
left parenthesis straight a right parenthesis space ab less than straight c squared over 4
left parenthesis straight b right parenthesis space ab greater or equal than straight c squared over 4
left parenthesis straight c right parenthesis space ab space greater or equal than straight c over 4
left parenthesis straight d right parenthesis space none space of space these end style

Solution 2

begin mathsize 12px style Correct space option colon space left parenthesis straight b right parenthesis
straight f left parenthesis straight x right parenthesis equals ax plus straight b over straight x
rightwards double arrow straight f apostrophe left parenthesis straight x right parenthesis equals straight a minus straight b over straight x squared
straight f apostrophe left parenthesis straight x right parenthesis equals 0
straight a minus straight b over straight x squared equals 0
rightwards double arrow straight x equals plus-or-minus square root of straight b over straight a end root
straight f apostrophe apostrophe open parentheses straight x close parentheses equals fraction numerator 2 straight b over denominator straight x cubed end fraction
straight f apostrophe apostrophe open parentheses square root of straight b over straight a end root close parentheses equals fraction numerator 2 straight b over denominator open parentheses square root of straight b over straight a end root close parentheses cubed end fraction greater than 0
rightwards double arrow straight x equals square root of straight b over straight a end root space has space straight a space minima.
straight f open parentheses square root of straight b over straight a end root close parentheses equals 2 square root of ab greater or equal than straight c
straight c over 2 less or equal than square root of ab
rightwards double arrow straight c squared over 4 less or equal than ab end style

Question 3

begin mathsize 12px style The space minimum space value space of space fraction numerator straight x over denominator log subscript straight e straight x end fraction space is
left parenthesis straight a right parenthesis space straight e
left parenthesis straight b right parenthesis space 1 divided by straight e
left parenthesis straight c right parenthesis space 1
left parenthesis straight d right parenthesis space none space of space these end style

Solution 3

 begin mathsize 12px style Correct space option colon thin space left parenthesis straight a right parenthesis
straight f open parentheses straight x close parentheses equals fraction numerator straight x over denominator log subscript straight e straight x end fraction
rightwards double arrow straight f apostrophe open parentheses straight x close parentheses equals fraction numerator log subscript straight e straight x minus 1 over denominator open parentheses log subscript straight e straight x close parentheses squared end fraction
straight f apostrophe open parentheses straight x close parentheses equals 0 space to space find space maxima space or space minima.
fraction numerator log subscript straight e straight x minus 1 over denominator open parentheses log subscript straight e straight x close parentheses squared end fraction equals 0
log subscript straight e straight x minus 1 equals 0
straight x equals straight e
rightwards double arrow straight f apostrophe apostrophe open parentheses straight x close parentheses equals fraction numerator negative 1 over denominator straight x open parentheses log subscript straight e straight x close parentheses squared end fraction plus fraction numerator 2 over denominator straight x open parentheses log subscript straight e straight x close parentheses cubed end fraction
rightwards double arrow straight f apostrophe apostrophe open parentheses straight e close parentheses equals 1 over straight e greater than 0
straight x equals straight e space is space straight a space minima.
thin space Minimum space value space of space straight f left parenthesis straight x right parenthesis equals straight e end style

Question 4

begin mathsize 12px style For space the space function space straight f left parenthesis straight x right parenthesis space equals space straight x space plus space 1 over straight x
left parenthesis straight a right parenthesis space straight x equals 1 space is space straight a space point space of space maximum
left parenthesis straight b right parenthesis space straight x equals negative 1 space is space straight a space point space of space minimum
left parenthesis straight c right parenthesis space maximumm space value greater than minimum space value
left parenthesis straight d right parenthesis space maximum space value space less than minimum space value end style

Solution 4

begin mathsize 12px style Correct space option colon thin space left parenthesis straight d right parenthesis
straight f left parenthesis straight x right parenthesis equals straight x plus 1 over straight x
rightwards double arrow straight f apostrophe open parentheses straight x close parentheses equals 1 minus 1 over straight x squared
To space find space maxima space or space minima comma
straight f apostrophe open parentheses straight x close parentheses equals 0
1 minus 1 over straight x squared equals 0
straight x equals plus-or-minus 1
straight f apostrophe apostrophe open parentheses straight x close parentheses equals 2 over straight x cubed
rightwards double arrow straight f apostrophe apostrophe open parentheses 1 close parentheses equals 2 greater than 0
function space has space minima space at space straight x equals 1
And
straight f apostrophe apostrophe open parentheses negative 1 close parentheses equals negative 2 less than 0
function space has space maxima space at space straight x equals negative 1
rightwards double arrow straight f left parenthesis 1 right parenthesis equals 2 space and space straight f open parentheses negative 1 close parentheses equals negative 2
rightwards double arrow Maximum space value less than space minimum space value. end style

Question 5

begin mathsize 12px style Let space straight f left parenthesis straight x right parenthesis equals straight x cubed plus 3 straight x squared minus 9 straight x plus 2. thin space Then comma space straight f left parenthesis straight x right parenthesis space has
left parenthesis straight a right parenthesis space straight a space maximum space at space straight x space equals space 1
left parenthesis straight b right parenthesis space straight a space minimum space at space straight x space equals space 1
left parenthesis straight c right parenthesis space neither space straight a space maximum space nor space straight a space minimum space at space straight x equals space minus 3
left parenthesis straight d right parenthesis space none space of space these end style

Solution 5

begin mathsize 12px style Correct space option colon left parenthesis straight b right parenthesis
straight f left parenthesis straight x right parenthesis equals straight x cubed plus 3 straight x squared minus 9 straight x plus 2
rightwards double arrow straight f apostrophe left parenthesis straight x right parenthesis equals 3 straight x squared plus 6 straight x minus 9
local space minima space or space maxima space must space have
straight f apostrophe left parenthesis straight x right parenthesis equals 0
3 straight x squared plus 6 straight x minus 9 equals 0
straight x squared plus 2 straight x minus 3 equals 0
rightwards double arrow open parentheses straight x plus 3 close parentheses open parentheses straight x minus 1 close parentheses equals 0
rightwards double arrow straight x equals negative 3 space or space straight x equals 1
straight f apostrophe apostrophe left parenthesis straight x right parenthesis equals 6 straight x plus 6
rightwards double arrow straight f apostrophe apostrophe left parenthesis negative 3 right parenthesis equals negative 12 less than 0
at space straight x equals negative 3 space local space maxima.
straight f apostrophe apostrophe left parenthesis 1 right parenthesis equals 12 greater than 0
at space straight x equals 1 space local space minima.
end style

Question 6

begin mathsize 12px style The space minimum space value space of space straight f left parenthesis straight x right parenthesis space equals space straight x to the power of 4 space minus straight x squared minus 2 straight x plus 6 space is
left parenthesis straight a right parenthesis space 6
left parenthesis straight b right parenthesis 4
left parenthesis straight c right parenthesis space 8
left parenthesis straight d right parenthesis space none space of space these end style

Solution 6

begin mathsize 12px style Correct space option colon left parenthesis straight b right parenthesis
straight f open parentheses straight x close parentheses equals straight x to the power of 4 minus straight x squared minus 2 straight x plus 6
rightwards double arrow straight f apostrophe open parentheses straight x close parentheses equals 4 straight x cubed minus 2 straight x minus 2
straight f apostrophe open parentheses straight x close parentheses equals 0
4 straight x cubed minus 2 straight x minus 2 equals 0
open parentheses straight x minus 1 close parentheses open parentheses 4 straight x squared plus 4 straight x plus 2 close parentheses equals 0
rightwards double arrow straight x equals 1 space
straight f apostrophe apostrophe open parentheses straight x close parentheses equals 12 straight x squared minus 2
straight f apostrophe apostrophe open parentheses 1 close parentheses equals 10 greater than 0
local space minima space at space straight x equals 1
rightwards double arrow straight f open parentheses 1 close parentheses equals 4

end style

Question 7

begin mathsize 12px style The space number space which space exceeds space its space sqaure space by space the space greatest space possible space quantity space is
left parenthesis straight a right parenthesis space 1 half
left parenthesis straight b right parenthesis space 1 fourth
left parenthesis straight c right parenthesis space 3 over 4
left parenthesis straight d right parenthesis space none space of space these end style

Solution 7

begin mathsize 12px style Correct space option colon space left parenthesis straight a right parenthesis
Let comma space the space number space be space straight x.
straight f left parenthesis straight x right parenthesis equals straight x minus straight x squared
straight f apostrophe open parentheses straight x close parentheses equals 1 minus 2 straight x
to space find space maxima space or space minima
straight f apostrophe open parentheses straight x close parentheses equals 0
1 minus 2 straight x equals 0
rightwards double arrow straight x equals 1 half
straight f apostrophe apostrophe open parentheses straight x close parentheses equals negative 2 less than 0
function space has space maxima space at space straight x equals 1 half. end style

Question 8

begin mathsize 12px style Let space straight f left parenthesis straight x right parenthesis space equals space open parentheses straight x minus straight a close parentheses squared plus open parentheses straight x minus straight b close parentheses squared plus open parentheses straight x minus straight c close parentheses squared. space Then space straight f left parenthesis straight x right parenthesis space has space straight a space minimum space at space straight x space equals space
left parenthesis straight a right parenthesis space fraction numerator straight a plus straight b plus straight c over denominator 3 end fraction
left parenthesis straight b right parenthesis space 3 square root of abc
left parenthesis straight c right parenthesis space fraction numerator 3 over denominator begin display style 1 over straight a plus 1 over straight b plus 1 over straight c end style end fraction
left parenthesis straight d right parenthesis space none space of space these end style

Solution 8

begin mathsize 12px style Correc space straight t space option colon space left parenthesis straight a right parenthesis
straight f left parenthesis straight x right parenthesis equals open parentheses straight x minus straight a close parentheses squared plus open parentheses straight x minus straight b close parentheses squared plus open parentheses straight x minus straight c close parentheses squared
rightwards double arrow straight f apostrophe left parenthesis straight x right parenthesis equals 2 open parentheses straight x minus straight a close parentheses plus 2 left parenthesis straight x minus straight b right parenthesis plus 2 left parenthesis straight x minus straight c right parenthesis
to space find space minima space or space maxima
straight f apostrophe left parenthesis straight x right parenthesis equals 0
2 open parentheses straight x minus straight a close parentheses plus 2 left parenthesis straight x minus straight b right parenthesis plus 2 left parenthesis straight x minus straight c right parenthesis equals 0
rightwards double arrow straight x equals fraction numerator straight a plus straight b plus straight c over denominator 3 end fraction
straight f apostrophe apostrophe left parenthesis straight x right parenthesis equals 6 greater than 0
function space has space minima space at space straight x equals fraction numerator straight a plus straight b plus straight c over denominator 3 end fraction. end style

Question 9

begin mathsize 12px style The space sum space of space two space non minus zero space numbers space is space 8 comma space the space minimum space value space of space the space sum space of space their space recirprocals space is
left parenthesis straight a right parenthesis space 1 fourth
left parenthesis straight b right parenthesis space 1 half
left parenthesis straight c right parenthesis space 1 over 8
left parenthesis straight d right parenthesis space none space of space these end style

Solution 9

begin mathsize 12px style Correct space option colon thin space left parenthesis straight b right parenthesis
Let comma space the space numbers space whose space sum space is space 8 space are space 8 comma space 8 minus straight x.
Given space straight f left parenthesis straight x right parenthesis equals 1 over straight x plus fraction numerator 1 over denominator 8 minus straight x end fraction
rightwards double arrow straight f apostrophe left parenthesis straight x right parenthesis equals fraction numerator negative 1 over denominator straight x squared end fraction plus 1 over open parentheses 8 minus straight x close parentheses squared
to space find space minima space or space maxima
straight f apostrophe left parenthesis straight x right parenthesis equals 0
rightwards double arrow fraction numerator negative 1 over denominator straight x squared end fraction plus 1 over open parentheses 8 minus straight x close parentheses squared equals 0
rightwards double arrow straight x equals 4
straight f apostrophe apostrophe left parenthesis straight x right parenthesis equals 2 over straight x cubed minus 2 over open parentheses 8 minus straight x close parentheses cubed
rightwards double arrow straight f apostrophe apostrophe left parenthesis 4 right parenthesis equals 2 over 4 cubed minus 2 over open parentheses 8 minus 4 close parentheses cubed equals 0
Minimum space value space of space the space sum space of space their space reciprocals equals 1 fourth plus 1 fourth equals 1 half
end style

Question 10

begin mathsize 12px style The space function space straight f left parenthesis straight x right parenthesis space equals space sum from straight r equals 1 to 5 of open parentheses straight x minus straight r close parentheses squared space assumes space minimum space value space at space straight x space equals space
left parenthesis straight a right parenthesis space 5
left parenthesis straight b right parenthesis space 5 divided by 2
left parenthesis straight c right parenthesis space 3
left parenthesis straight d right parenthesis space 2 end style

Solution 10

begin mathsize 12px style Correct space option colon left parenthesis straight c right parenthesis
straight f left parenthesis straight x right parenthesis equals begin inline style sum from straight r equals 1 to 5 of end style open parentheses straight x minus straight r close parentheses squared
straight f left parenthesis straight x right parenthesis equals open parentheses straight x minus 1 close parentheses squared plus open parentheses straight x minus 2 close parentheses squared plus open parentheses straight x minus 3 close parentheses squared plus open parentheses straight x minus 4 close parentheses squared plus open parentheses straight x minus 5 close parentheses squared
rightwards double arrow straight f apostrophe left parenthesis straight x right parenthesis equals 2 open parentheses straight x minus 1 plus straight x minus 2 plus straight x minus 3 plus straight x minus 4 plus straight x minus 5 close parentheses
rightwards double arrow straight f apostrophe left parenthesis straight x right parenthesis equals 2 open parentheses 5 straight x minus 15 close parentheses
straight f apostrophe left parenthesis straight x right parenthesis equals 0
2 open parentheses 5 straight x minus 15 close parentheses equals 0
rightwards double arrow straight x equals 3
straight f apostrophe apostrophe left parenthesis straight x right parenthesis equals 10 greater than 0
function space has space minima space at space straight x equals 3
end style

Question 11

begin mathsize 12px style At space straight x equals fraction numerator 5 straight pi over denominator 6 end fraction comma straight f left parenthesis straight x right parenthesis equals 2 space sin space 3 straight x plus 3 space cos space 3 straight x space is
left parenthesis straight a right parenthesis space 0
left parenthesis straight b right parenthesis space maximum
left parenthesis straight c right parenthesis space minimum
left parenthesis straight d right parenthesis space none space of space these end style

Solution 11

begin mathsize 12px style Correct space option colon space left parenthesis straight d right parenthesis
straight f left parenthesis straight x right parenthesis equals 2 sin 3 straight x plus 3 cos 3 straight x
rightwards double arrow straight f apostrophe left parenthesis straight x right parenthesis equals 6 cos 3 straight x minus 9 sin 3 straight x
to space find space maxima space or space minima
straight f apostrophe left parenthesis straight x right parenthesis equals 0
6 cos 3 straight x minus 9 sin 3 straight x equals 0
rightwards double arrow tan 3 straight x equals 2 over 3
straight f apostrophe open parentheses fraction numerator 5 straight pi over denominator 6 end fraction close parentheses equals tan open parentheses 3 cross times fraction numerator 5 straight pi over denominator 6 end fraction close parentheses
straight f apostrophe open parentheses fraction numerator 5 straight pi over denominator 6 end fraction close parentheses equals tan open parentheses fraction numerator 5 straight pi over denominator 2 end fraction close parentheses
rightwards double arrow straight f apostrophe open parentheses fraction numerator 5 straight pi over denominator 6 end fraction close parentheses equals tan open parentheses 2 straight pi plus straight pi over 2 close parentheses
rightwards double arrow straight f apostrophe open parentheses fraction numerator 5 straight pi over denominator 6 end fraction close parentheses equals tan open parentheses straight pi over 2 close parentheses space which space is space not space defined.
Hence comma space straight x equals fraction numerator 5 straight pi over denominator 6 end fraction space is space not space straight a space critical space point.
end style

Question 12

begin mathsize 12px style If space straight x space lies space in space the space interval space left square bracket 0 comma 1 right square bracket comma space then space the space least space value space of space straight x squared plus straight x plus 1 space is
left parenthesis straight a right parenthesis space 3
left parenthesis straight b right parenthesis space 3 divided by 4
left parenthesis straight c right parenthesis space 1
left parenthesis straight d right parenthesis space none space of space these end style

Solution 12

begin mathsize 12px style Correct space option colon thin space left parenthesis straight c right parenthesis
straight f left parenthesis straight x right parenthesis equals straight x squared plus straight x plus 1
straight f apostrophe left parenthesis straight x right parenthesis equals 2 straight x plus 1
To space find space minima space
straight f apostrophe left parenthesis straight x right parenthesis equals 0
2 straight x plus 1 equals 0
rightwards double arrow straight x equals fraction numerator negative 1 over denominator 2 end fraction space not an element of open square brackets 0 comma 1 close square brackets
straight f left parenthesis 0 right parenthesis equals 0 space and space straight f left parenthesis 1 right parenthesis equals 3 greater than 0
Hence comma space at space straight x equals 1 space function space has space minima. end style

Question 13

begin mathsize 12px style The space least space value space of space the space fuction space straight f left parenthesis straight x right parenthesis space equals space straight x cubed space minus space 18 straight x squared space plus space 96 straight x space in space the space interval space left square bracket 0 comma 9 right square bracket space is
left parenthesis straight a right parenthesis space 126
left parenthesis straight b right parenthesis space 135
left parenthesis straight c right parenthesis space 160
left parenthesis straight d right parenthesis space 0 end style

Solution 13

begin mathsize 12px style Correct space option colon space left parenthesis straight d right parenthesis
straight f left parenthesis straight x right parenthesis equals straight x cubed minus 18 straight x squared plus 96 straight x
rightwards double arrow straight f apostrophe left parenthesis straight x right parenthesis equals 3 straight x squared minus 36 straight x plus 96
To space find space least space value space of space the space function comma
straight f apostrophe left parenthesis straight x right parenthesis equals 0
3 straight x squared minus 36 straight x plus 96 equals 0
straight x squared minus 12 straight x plus 32 equals 0
rightwards double arrow open parentheses straight x minus 4 close parentheses open parentheses straight x minus 8 close parentheses equals 0
straight x equals 4 space or space straight x equals 8
straight f open parentheses 0 close parentheses equals 0
straight f open parentheses 9 close parentheses equals 135
straight f left parenthesis 4 right parenthesis equals 160
straight f left parenthesis 8 right parenthesis equals 128
at space straight x equals 0 space function space has space minimum space value space on space open square brackets 0 comma 9 close square brackets
end style

Question 14

begin mathsize 12px style The space maximum space value space of space straight f left parenthesis straight x right parenthesis space equals space fraction numerator straight x over denominator 4 minus straight x plus straight x squared end fraction space on space left square bracket negative 1 comma 1 right square bracket space is
left parenthesis straight a right parenthesis space minus 1 fourth
left parenthesis straight b right parenthesis space minus 1 third
left parenthesis straight c right parenthesis space 1 over 6
left parenthesis straight d right parenthesis space 1 fifth end style

Solution 14

begin mathsize 12px style straight f left parenthesis straight x right parenthesis equals fraction numerator straight x over denominator 4 minus straight x plus straight x squared end fraction
rightwards double arrow straight f apostrophe open parentheses straight x close parentheses equals fraction numerator 4 minus straight x squared over denominator open parentheses 4 minus straight x plus straight x squared close parentheses squared end fraction
To space find space minimum space or space maximum space value
straight f apostrophe open parentheses straight x close parentheses equals 0
fraction numerator 4 minus straight x squared over denominator open parentheses 4 minus straight x plus straight x squared close parentheses squared end fraction equals 0
4 minus straight x squared equals 0
straight x equals plus-or-minus 2 not an element of open square brackets negative 1 comma 1 close square brackets
straight f open parentheses negative 1 close parentheses equals fraction numerator negative 1 over denominator 6 end fraction space and space space straight f open parentheses 1 close parentheses equals 1 fourth
Hence comma space maximum space value space is space 1 fourth.

NOTE colon space Options space in space the space book space are space not space matching space with space the space solution. space
Above space solution space is space based space on space the space question space given space in space the space book. end style

Question 15

begin mathsize 12px style The space point space on space the space curve space straight y squared space equals space 4 straight x space which space is space nearest space to space the space point space left parenthesis 2 comma 1 right parenthesis space is
left parenthesis straight a right parenthesis space open parentheses 1 comma 2 square root of 2 close parentheses
left parenthesis straight b right parenthesis space open parentheses 1 comma 2 close parentheses
left parenthesis straight c right parenthesis space open parentheses 1 comma negative 2 close parentheses
left parenthesis straight b right parenthesis space open parentheses negative 2 comma 1 close parentheses end style

Solution 15

begin mathsize 12px style Correct space option colon thin space left parenthesis straight b right parenthesis
straight y squared equals 4 straight x rightwards double arrow straight x equals straight y squared over 4
rightwards double arrow straight d equals square root of open parentheses straight x minus 2 close parentheses squared plus open parentheses straight y minus 1 close parentheses squared end root
rightwards double arrow straight d squared equals open parentheses straight x minus 2 close parentheses squared plus open parentheses straight y minus 1 close parentheses squared
rightwards double arrow straight d squared equals open parentheses straight y squared over 4 minus 2 close parentheses squared plus open parentheses straight y minus 1 close parentheses squared
Let comma space straight u equals equals open parentheses straight y squared over 4 minus 2 close parentheses squared plus open parentheses straight y minus 1 close parentheses squared
rightwards double arrow du over dy equals 2 open parentheses straight y squared over 4 minus 2 close parentheses straight y over 2 plus 2 open parentheses straight y minus 1 close parentheses
To space find space minima
du over dy equals 0
2 open parentheses straight y squared over 4 minus 2 close parentheses straight y over 2 plus 2 open parentheses straight y minus 1 close parentheses equals 0
rightwards double arrow straight y equals 2 rightwards double arrow straight x equals 1 space space space space space space space space open parentheses straight x equals straight y squared over 4 close parentheses
fraction numerator straight d squared straight u over denominator dy squared end fraction equals fraction numerator 3 straight y squared over denominator 4 end fraction
rightwards double arrow open parentheses fraction numerator straight d squared straight u over denominator dy squared end fraction close parentheses subscript open parentheses 1 comma 2 close parentheses end subscript equals 3 greater than 0
Hence comma space nearest space point space is space left parenthesis 1 comma 2 right parenthesis. end style

Question 16

begin mathsize 12px style If space straight x plus straight y equals 8 comma space then space the space maximum space value space of space xy space is
left parenthesis straight a right parenthesis space 8
left parenthesis straight b right parenthesis space 16
left parenthesis straight c right parenthesis space 20
left parenthesis straight d right parenthesis space 24 end style

Solution 16

begin mathsize 12px style Correct space option colon thin space left parenthesis straight b right parenthesis
Given space straight x plus straight y equals 8
rightwards double arrow straight y equals 8 minus straight x
To space find space the space maximum space value space of space xy comma
rightwards double arrow straight Y equals straight x open parentheses 8 minus straight x close parentheses
straight Y equals 8 straight x minus straight x squared
dY over dx equals 8 minus 2 straight x
for space maximum space or space minimum space value space we space must space have space
dY over dx equals 0
8 minus 2 straight x equals 0 rightwards double arrow straight x equals 4
rightwards double arrow straight y equals 4
fraction numerator straight d squared straight Y over denominator dx squared end fraction equals negative 2 less than 0
Hence comma space maximum space value space of space xy space is space 16.
end style

Question 17

begin mathsize 12px style The space least space and space gretest space values space of space straight f left parenthesis straight x right parenthesis equals straight x cubed minus 6 straight x squared plus 9 straight x space in space left square bracket 0 comma 6 right square bracket comma space are
left parenthesis straight a right parenthesis space 3 comma 4
left parenthesis straight b right parenthesis space 0 comma 6
left parenthesis straight c right parenthesis space 0 comma 3
left parenthesis straight d right parenthesis space 3 comma 6 right parenthesis end style

Solution 17

begin mathsize 12px style straight f left parenthesis straight x right parenthesis equals straight x cubed minus 6 straight x squared plus 9 straight x
rightwards double arrow straight f apostrophe left parenthesis straight x right parenthesis equals 3 straight x squared minus 12 straight x plus 9
To space find space minimum space or space maximum space value space we space have
straight f apostrophe left parenthesis straight x right parenthesis equals 0
3 straight x squared minus 12 straight x plus 9 equals 0
rightwards double arrow straight x squared minus 4 straight x plus 3 equals 0
rightwards double arrow straight x equals 1 space or space straight x equals 3
straight f left parenthesis 0 right parenthesis equals 0
straight f left parenthesis 6 right parenthesis equals 54
straight f left parenthesis 1 right parenthesis equals 4
straight f left parenthesis 3 right parenthesis equals 0
Least space value space and space greatest space value space of space the space functions space are space 0 space and space 54 space respectively.


NOTE colon space Options space are space not space matching space with space the space solution. space
Above space solution space is space according space to space the space question space given space in space the space book. end style

Question 18

begin mathsize 12px style straight f left parenthesis straight x right parenthesis space equals space sin space straight x plus square root of 3 space cos space straight x space is space maximum space when space straight x space equals space
left parenthesis straight a right parenthesis space straight pi over 3
left parenthesis straight b right parenthesis space straight pi over 4
left parenthesis straight c right parenthesis space straight pi over 6
left parenthesis straight d right parenthesis space 0 end style

Solution 18

begin mathsize 12px style Correct space option colon thin space left parenthesis straight c right parenthesis
straight f left parenthesis straight x right parenthesis space equals space sin space straight x plus square root of 3 space cos space straight x
rightwards double arrow straight f apostrophe left parenthesis straight x right parenthesis space equals cosx minus square root of 3 space sinx
for space maxima space or space minima comma
straight f apostrophe left parenthesis straight x right parenthesis equals 0
cosx minus square root of 3 space sinx equals 0
rightwards double arrow tanx equals fraction numerator 1 over denominator square root of 3 end fraction rightwards double arrow straight x equals straight pi over 6
straight f apostrophe apostrophe left parenthesis straight x right parenthesis equals minus sinx minus square root of 3 cosx
rightwards double arrow straight f apostrophe apostrophe open parentheses straight pi over 6 close parentheses equals negative sin straight pi over 6 minus square root of 3 cos straight pi over 6 equals fraction numerator negative 1 minus square root of 3 over denominator 2 end fraction less than 0
function space has space local space maima space at space straight x equals straight pi over 6 space end style

Question 19

begin mathsize 12px style if space straight a space cone space of space maximum space volume space is space inscribed space in space straight a space given space sphere comma space then space the space ratio space of space the space height space of
of space the space cone space to space the space diameter space of space the space sphere space is
left parenthesis straight a right parenthesis space 3 divided by 4
left parenthesis straight b right parenthesis space 1 divided by 3
left parenthesis straight c right parenthesis space 1 divided by 4
left parenthesis straight d right parenthesis space 2 divided by 3 end style

Solution 19

begin mathsize 12px style Correct space option colon space left parenthesis straight d right parenthesis
Volume space of space straight a space cone equals 1 third πr squared straight h
where space straight R space radius space of space sphere space and space straight r space be space the space radius space of space the space base.
Given space that space straight h equals straight R plus square root of straight R squared minus straight r squared end root
rightwards double arrow straight h minus straight R equals square root of straight R squared minus straight r squared end root
rightwards double arrow straight r squared equals 2 hr minus straight h squared
rightwards double arrow straight V equals 1 third straight pi open parentheses 2 hr minus straight h squared close parentheses straight h
rightwards double arrow straight V equals straight pi over 3 open parentheses 2 straight h squared straight R minus straight h cubed close parentheses
rightwards double arrow dV over dh equals straight pi over 3 open parentheses 4 hR minus 3 straight h squared close parentheses
To space find space maximum space or space minimum space values space of space straight V comma
straight pi over 3 open parentheses 4 hR minus 3 straight h squared close parentheses equals 0
rightwards double arrow straight h equals fraction numerator 4 straight R over denominator 3 end fraction
fraction numerator straight d squared straight V over denominator dh squared end fraction equals straight pi over 3 open parentheses 4 straight R minus 6 straight h close parentheses
rightwards double arrow open parentheses fraction numerator straight d squared straight V over denominator dh squared end fraction close parentheses subscript straight h equals fraction numerator 4 straight R over denominator 3 end fraction end subscript equals negative fraction numerator 4 πR over denominator 3 end fraction less than 0
Volume space is space maximum space at space straight h equals fraction numerator 4 straight R over denominator 3 end fraction
Diameter space is space straight d equals 2 straight R
rightwards double arrow space straight h equals fraction numerator 2 cross times 2 straight R over denominator 3 end fraction
rightwards double arrow fraction numerator straight h over denominator 2 straight R end fraction equals 2 over 3

dV over dh equals end style

Question 20

begin mathsize 12px style The space minimum space value space of space open parentheses straight x squared plus 250 over straight x close parentheses space is
left parenthesis straight a right parenthesis space 75
left parenthesis straight b right parenthesis space 50
left parenthesis straight c right parenthesis space 25
left parenthesis straight d right parenthesis 55 space end style

Solution 20

begin mathsize 12px style Correct space option colon left parenthesis straight a right parenthesis
straight f left parenthesis straight x right parenthesis equals straight x squared plus 250 over straight x
rightwards double arrow straight f apostrophe open parentheses straight x close parentheses equals 2 straight x minus 250 over straight x squared
For space the space local space minim space straight a space or space maxima space we space must space have
straight f apostrophe open parentheses straight x close parentheses equals 0
2 straight x minus 250 over straight x squared equals 0
rightwards double arrow straight x equals 5
straight f apostrophe apostrophe left parenthesis straight x right parenthesis equals 2 plus 500 over straight x cubed
straight f apostrophe apostrophe left parenthesis 5 right parenthesis equals 2 plus 500 over 125 greater than 0
function space has space minima space at space straight x equals 5.
straight f open parentheses 5 close parentheses equals 75. end style

Question 21

begin mathsize 12px style if space straight f left parenthesis straight x right parenthesis space equals space straight x space plus space fraction numerator 1 over denominator space straight x end fraction comma straight x greater than 0 comma space then space its space greatest space value space is
left parenthesis straight a right parenthesis space minus 2
left parenthesis straight b right parenthesis space 0
left parenthesis straight c right parenthesis space 3
left parenthesis straight d right parenthesis space none space of space these end style

Solution 21

begin mathsize 12px style Correct space option colon left parenthesis straight d right parenthesis
straight f left parenthesis straight x right parenthesis equals straight x plus 1 over straight x
rightwards double arrow straight f apostrophe open parentheses straight x close parentheses equals 1 minus 1 over straight x squared
For space minimum space or space maximum space value space of space the space function
straight f apostrophe open parentheses straight x close parentheses equals 0
rightwards double arrow 1 minus 1 over straight x squared equals 0
rightwards double arrow straight x equals plus-or-minus 1
straight f apostrophe apostrophe left parenthesis straight x right parenthesis equals 2 over straight x cubed
rightwards double arrow straight f apostrophe apostrophe left parenthesis 1 right parenthesis equals 1 greater than 0 rightwards double arrow function space has space minima space at space straight x equals 1.
straight f apostrophe apostrophe left parenthesis negative 1 right parenthesis equals negative 1 greater than 0 rightwards double arrow function space has space minima space at space straight x equals negative 1. end style

Question 22

begin mathsize 12px style if space straight f left parenthesis straight x right parenthesis space equals space fraction numerator 1 over denominator 4 straight x squared plus 2 straight x plus 1 end fraction comma space then space its space maximum space value space is
left parenthesis straight a right parenthesis space 4 over 3
left parenthesis straight b right parenthesis space 2 over 3
left parenthesis straight c right parenthesis space 1
left parenthesis straight d right parenthesis space 3 over 4 end style

Solution 22

begin mathsize 12px style Correct space option space left parenthesis straight a right parenthesis
straight f left parenthesis straight x right parenthesis equals fraction numerator 1 over denominator 4 straight x squared plus 2 straight x plus 1 end fraction
rightwards double arrow straight f apostrophe open parentheses straight x close parentheses equals 8 straight x plus 2
For space local space minima space or space maxima space we space have
straight f apostrophe open parentheses straight x close parentheses equals 8 straight x plus 2 equals 0
rightwards double arrow straight x equals fraction numerator negative 1 over denominator 4 end fraction
straight f apostrophe apostrophe open parentheses straight x close parentheses equals 8 greater than 0
rightwards double arrow function space has space maxima space at space straight x equals fraction numerator negative 1 over denominator 4 end fraction
straight f open parentheses fraction numerator negative 1 over denominator 4 end fraction close parentheses equals 4 over 3 end style

Question 23

begin mathsize 12px style Let space straight x comma straight y space be space two space variables space and space straight x greater than 0 comma space xy space equals space 1 space then space minimum space value space of space straight x plus straight y space is
left parenthesis straight a right parenthesis space 1
left parenthesis straight b right parenthesis space 2
left parenthesis straight c right parenthesis space 2 1 half
left parenthesis straight d right parenthesis space 3 1 third end style

Solution 23

begin mathsize 12px style Correct space option colon thin space left parenthesis straight b right parenthesis
Given space xy equals 1. space To space find space minimum space value space of space straight x plus straight y
rightwards double arrow straight y equals 1 over straight x
straight f left parenthesis straight x right parenthesis equals straight x plus 1 over straight x
rightwards double arrow straight f apostrophe left parenthesis straight x right parenthesis equals 1 minus 1 over straight x squared
To space find space local space maxima space or space minima space we space have
straight f apostrophe left parenthesis straight x right parenthesis equals 0
1 minus 1 over straight x squared equals 0
rightwards double arrow x equals plus-or-minus 1 rightwards double arrow y equals plus-or-minus 1
But space given space that space straight x greater than 0 rightwards double arrow straight x equals 1 comma space straight y equals 1
straight f apostrophe apostrophe left parenthesis straight x right parenthesis equals 2 over straight x cubed
straight f apostrophe apostrophe left parenthesis 1 right parenthesis equals 2 greater than 0
function space has space minima space at space straight x equals 1
straight f left parenthesis 1 right parenthesis equals 2. end style

Question 24

begin mathsize 12px style straight f left parenthesis straight x right parenthesis space equals space 1 plus 2 space sin space straight x plus 3 space cos squared straight x comma 0 less or equal than straight x less or equal than fraction numerator 2 straight pi over denominator 3 end fraction space is
left parenthesis straight a right parenthesis space Minimum space at space straight x space equals space straight pi divided by 2
left parenthesis straight b right parenthesis space Maximum space at space straight x space equals space sin to the power of negative 1 end exponent space left parenthesis 1 divided by square root of 3 right parenthesis
left parenthesis straight c right parenthesis space Minimum space at space straight x space equals space straight pi divided by 6
left parenthesis straight d right parenthesis space Maximum space at space sin to the power of negative 1 end exponent space left parenthesis 1 divided by 6 right parenthesis end style

Solution 24

begin mathsize 12px style Correct space option colon space left parenthesis straight a right parenthesis
straight f left parenthesis straight x right parenthesis equals 1 plus 2 sinx plus 3 cos squared straight x
rightwards double arrow straight f apostrophe left parenthesis straight x right parenthesis equals 2 cosx minus 6 cosxsinx
rightwards double arrow straight f apostrophe left parenthesis straight x right parenthesis equals 2 cosx minus 3 sin 2 straight x
to space find space minima space or space maxima space of space the space function
2 cosx minus 6 cosxsinx equals 0
2 cosx open parentheses 1 minus 3 sinx close parentheses equals 0
rightwards double arrow cosx equals 0 space or space sinx equals 1 third
straight x equals straight pi over 2 space space or space space straight x equals sin to the power of negative 1 end exponent open parentheses 1 third close parentheses
straight f apostrophe apostrophe left parenthesis straight x right parenthesis equals negative 2 sinx minus 6 cos 2 straight x
straight f apostrophe apostrophe open parentheses straight pi over 2 close parentheses equals 4 greater than 0
rightwards double arrow straight x equals straight pi over 2 space is space straight a space local space minima.
straight f apostrophe apostrophe open parentheses sin to the power of negative 1 end exponent open parentheses 1 third close parentheses close parentheses equals negative open parentheses 2 over 3 plus 4 square root of 2 close parentheses less than 0
function space has space maxima space at space straight x equals sin to the power of negative 1 end exponent open parentheses 1 third close parentheses space
end style

Question 25

begin mathsize 12px style The space function space straight f left parenthesis straight x right parenthesis space equals space 2 straight x cubed minus 15 straight x squared plus 36 straight x plus 4 space is space maximum space at space straight x space equals space
left parenthesis straight a right parenthesis space 3
left parenthesis straight b right parenthesis space 0
left parenthesis straight c right parenthesis space 4
left parenthesis straight d right parenthesis space 2 end style

Solution 25

begin mathsize 12px style Correct space option colon space left parenthesis straight d right parenthesis space
straight f left parenthesis straight x right parenthesis space equals space 2 straight x cubed minus 15 straight x squared plus 36 straight x plus 4
rightwards double arrow straight f apostrophe open parentheses straight x close parentheses equals 6 straight x squared minus 30 straight x plus 36
for space local space minima space or space maxima
straight f apostrophe open parentheses straight x close parentheses equals 0
6 straight x squared minus 30 straight x plus 36 equals 0
straight x squared minus 5 straight x plus 6 equals 0
rightwards double arrow straight x equals 2 space or space straight x equals 3
straight f apostrophe apostrophe left parenthesis straight x right parenthesis equals 12 straight x minus 30
straight f apostrophe apostrophe left parenthesis 2 right parenthesis equals negative 6 less than 0
straight x equals 2 space has space maxima.
straight f apostrophe apostrophe left parenthesis 3 right parenthesis equals 6 greater than 0
straight x equals 3 space has space minima.
end style

Question 26

begin mathsize 12px style The space maximum space value space of space straight f left parenthesis straight x right parenthesis space equals space fraction numerator straight x over denominator 4 plus straight x plus straight x squared end fraction space on space left square bracket negative 1 comma 1 right square bracket space is
left parenthesis straight a right parenthesis space minus 1 fourth
left parenthesis straight b right parenthesis space minus 1 third
left parenthesis straight c right parenthesis space 1 over 6
left parenthesis straight d right parenthesis space 1 fifth end style

Solution 26

begin mathsize 12px style Correct space option colon space left parenthesis straight c right parenthesis
straight f left parenthesis straight x right parenthesis equals fraction numerator straight x over denominator 4 plus straight x plus straight x squared end fraction
rightwards double arrow straight f apostrophe left parenthesis straight x right parenthesis equals fraction numerator 4 minus straight x squared over denominator open parentheses 4 plus straight x plus straight x squared close parentheses squared end fraction
For space straight a space local space maxima space or space minima comma
straight f apostrophe left parenthesis straight x right parenthesis equals 0
fraction numerator 4 minus straight x squared over denominator open parentheses 4 plus straight x plus straight x squared close parentheses squared end fraction equals 0
rightwards double arrow straight x equals plus-or-minus 2 not an element of open square brackets negative 1 comma 1 close square brackets
straight f left parenthesis 1 right parenthesis equals 1 over 6 greater than 0
straight f left parenthesis negative 1 right parenthesis equals fraction numerator negative 1 over denominator 4 end fraction less than 0
rightwards double arrow 1 over 6 space is space the space maximum space value.
end style

Question 27

Let f(x) = 2x3 - 3x2 - 12x + 5 on [-2,4]. The relative maximum occurs at x = 

(a) -2

(b) -1 

(c) 2

(d) 4

Solution 27

begin mathsize 12px style Correct space option colon thin space left parenthesis straight b right parenthesis
straight f left parenthesis straight x right parenthesis space equals space 2 straight x cubed space minus space 3 straight x squared space minus space 12 straight x space plus space 5
rightwards double arrow straight f apostrophe open parentheses straight x close parentheses equals 6 straight x squared minus 6 straight x minus 12
For space local space maxima space or space minima space we space have
straight f apostrophe left parenthesis straight x right parenthesis equals 0
6 straight x squared minus 6 straight x minus 12 equals 0
rightwards double arrow straight x squared minus straight x minus 2 equals 0
rightwards double arrow straight x equals 2 space or space straight x equals negative 1
straight f apostrophe apostrophe left parenthesis straight x right parenthesis equals 12 straight x minus 6
straight f apostrophe apostrophe left parenthesis 2 right parenthesis equals 18 greater than 0
function space has space local space minima space at space straight x equals 2.
straight f apostrophe apostrophe left parenthesis negative 1 right parenthesis equals negative 18 less than 0
function space has space local space maxima space at space straight x equals negative 1. end style

Question 28

begin mathsize 12px style The space minimum space value space of space straight x space log subscript straight e space straight x space is space equal space to
left parenthesis straight a right parenthesis space straight e
left parenthesis straight b right parenthesis space 1 divided by straight e
left parenthesis straight c right parenthesis space minus 1 divided by straight e
left parenthesis straight d right parenthesis space 2 straight e
left parenthesis straight e right parenthesis space minus straight e end style

Solution 28

begin mathsize 12px style Correct space option colon space left parenthesis straight c right parenthesis
straight f left parenthesis straight x right parenthesis equals xlog subscript straight e straight x
rightwards double arrow straight f apostrophe left parenthesis straight x right parenthesis equals 1 plus log subscript straight e straight x
to space find space maxima space or space minima
straight f apostrophe left parenthesis straight x right parenthesis equals 0
rightwards double arrow 1 plus log subscript straight e straight x equals 0
rightwards double arrow straight x equals 1 over straight e
straight f apostrophe apostrophe left parenthesis straight x right parenthesis equals 1 over straight x
straight f apostrophe apostrophe open parentheses 1 over straight e close parentheses equals straight e greater than 0
straight x equals 1 over straight e space is space straight a space local space minima.
rightwards double arrow Minimum space value space of space the space function space is
straight f open parentheses 1 over straight e close parentheses equals 1 over straight e log subscript straight e open parentheses 1 over straight e close parentheses equals fraction numerator negative 1 over denominator straight e end fraction end style

Question 29

begin mathsize 12px style The space minimum space value space of space the space function space straight f left parenthesis straight x right parenthesis space equals space 2 straight x cubed minus 21 straight x squared plus 36 straight x minus 20 space is
left parenthesis straight a right parenthesis space minus 128
left parenthesis straight b right parenthesis space minus 126
left parenthesis straight c right parenthesis space minus 120
left parenthesis straight d right parenthesis space none space of space these end style

Solution 29

begin mathsize 12px style Correct space option colon space left parenthesis straight a right parenthesis
straight f left parenthesis straight x right parenthesis space equals space 2 straight x cubed minus 21 straight x squared plus 36 straight x minus 20 space
rightwards double arrow straight f apostrophe left parenthesis straight x right parenthesis equals 6 straight x squared minus 42 straight x plus 36
For space local space maxima space or space minima space
6 straight x squared minus 42 straight x plus 36 equals 0
straight x squared minus 7 straight x plus 6 equals 0
rightwards double arrow straight x equals 1 space or space straight x equals 6
straight f apostrophe apostrophe left parenthesis straight x right parenthesis equals 12 straight x minus 42
rightwards double arrow straight f apostrophe apostrophe left parenthesis 1 right parenthesis equals negative 30 less than 0
also comma space straight f apostrophe apostrophe left parenthesis 6 right parenthesis equals 30 greater than 0
function space has space minima space at space straight x equals 6
rightwards double arrow straight f open parentheses 6 close parentheses equals negative 128 end style

Chapter 18 - Maxima and Minima Excercise Ex. 18VSAQ

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

W e space h a v e comma space f open parentheses x close parentheses equals x log subscript e x
T h u s comma space f open parentheses 1 over e close parentheses equals 1 over e log subscript e 1 over e
rightwards double arrow f open parentheses 1 over e close parentheses equals 1 over e log subscript e e to the power of minus 1 end exponent
rightwards double arrow f open parentheses 1 over e close parentheses equals 1 over e open parentheses minus 1 close parentheses
rightwards double arrow f open parentheses 1 over e close parentheses equals minus 1 over e
T h u s comma space t h e space p o i n t space w h e r e space f open parentheses x close parentheses space a t t a i n s space m i n i m u m space v a l u e space i s space open parentheses 1 over e comma minus 1 over e close parentheses

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10