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RD Sharma Solution for Class 12 Science Mathematics Chapter 18 - Maxima and Minima

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Our RD Sharma Textbook Solutions are considered extremely helpful for solving the tough questions which are asked in the CBSE Class 12 exam. TopperLearning Textbook Solutions are compiled by subject experts. Herein, you can find all the answers to the textbook questions for Chapter 18 - Maxima and Minima.

All our solutions are created in accordance with the latest CBSE syllabus, and they are amended from time to time to provide the most relevant answers. Our free RD Sharma Solutions for CBSE Class 12 Mathematics will strengthen your fundamentals in Mathematics and will help you in your attempts to score more marks in the final examination. CBSE Class 12 students can refer to our solutions any time — while doing their homework and while preparing for the exam.

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RD Sharma Solution for Class 12 Science Mathematics Chapter 18 - Maxima and Minima Page/Excercise 18.1

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

RD Sharma Solution for Class 12 Science Mathematics Chapter 18 - Maxima and Minima Page/Excercise 18.2

Solution 1

Solution 2


Solution 3


Solution 4


Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10


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

Solution 12

Solution 13

Solution 14

RD Sharma Solution for Class 12 Science Mathematics Chapter 18 - Maxima and Minima Page/Excercise 18.3

Solution 1(i)

Solution 1(ii)

Solution 1(iii)


Solution 1(iv)

Solution 1(v)

Solution 1(vi)

Solution 1(vii)

Solution 1(viii)

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

Solution 1(ix)

Solution 1(x)

Solution 1(xi)

Solution 1(xii)

Solution 2(i)

Solution 2(ii)

Solution 2(iii)

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

RD Sharma Solution for Class 12 Science Mathematics Chapter 18 - Maxima and Minima Page/Excercise 18.4

Solution 1(i)

Solution 1(ii)

Solution 1(iii)

Solution 1(iv)

Solution 2

Solution 3

Solution 4

Solution 5

RD Sharma Solution for Class 12 Science Mathematics Chapter 18 - Maxima and Minima Page/Excercise 18.5

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

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

Solution 9

Solution 10

Solution 11

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

Solution 13

Solution 14

Solution 15

Solution 16


Solution 17

Solution 18

Solution 19

Solution 20

Solution 21

 

 

Solution 22

 

 

Solution 23

Solution 24

Solution 25

Solution 26

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

Solution 28

Solution 29

Solution 30

Solution 31

Solution 32

Solution 33

Solution 34

Solution 35

Solution 36

Solution 37

Solution 38

Solution 39

Solution 40

Solution 41

Solution 42

Solution 43

Solution 44

Solution 45

RD Sharma Solution for Class 12 Science Mathematics Chapter 18 - Maxima and Minima Page/Excercise MCQ

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

RD Sharma Solution for Class 12 Science Mathematics Chapter 18 - Maxima and Minima Page/Excercise 18VSAQ

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

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

Solution 7

Solution 8

Solution 9

Solution 10

TopperLearning provides step-by-step solutions for each question in each chapter in the RD Sharma textbook for class 9. Access the CBSE Class 9 Mathematics Chapter 18 - Maxima and Minima for free. The textbook questions have been solved by our subject matter experts to help you understand how to answer them. Our RD Sharma Textbook Solutions will help you to study and revise, and you can easily clear your fundamentals of Chapter 18 - Maxima and Minima.

Text Book Solutions

CBSE XII Science - Mathematics

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