Ask a Doubt
Request a call back

Join NOW to get access to exclusive study material for best results

JEE Class main Answered

If f(x)f(x) is a differentiable function and g(x)g(x)is a double differentiable function such that |f(x)|≤1|f(x)|≤1 and f'(x)=g(x)f′(x)=g(x). If f2(0)+g2(0)=9f2(0)+g2(0)=9. Prove that there exists some c∈(–3,3)c∈(–3,3)such thatg(c).g''(c)<0
Asked by abhyudith | 06 Apr, 2020, 23:40: PM
answered-by-expert Expert Answer
If f(x) is a differentiable function and g(x)is a double differentiable function such that |f(x)|≤1 and f′(x)=g(x). If f2(0)+g2(0)=9. Prove that there exists some c∈(–3,3) such that g(c).g''(c)<0
Solution: 
TO prove: g(x).g''(x)<0 for some c∈(–3,3)
This means that one of g(x) or g"(x) is negative is negative
Let's assume that both are positive.
Since, |f(x)|≤1
rightwards double arrow f left parenthesis x right parenthesis element of left square bracket negative 1 comma space 1 right square bracket space rightwards double arrow space f squared left parenthesis x right parenthesis space less or equal than 1 space a n d space f squared left parenthesis 0 right parenthesis space plus space g squared left parenthesis 0 right parenthesis space equals space 9 rightwards double arrow g left parenthesis 0 right parenthesis less or equal than 9 G i v e n colon space f apostrophe left parenthesis x right parenthesis equals g left parenthesis x right parenthesis rightwards double arrow integral subscript negative 3 end subscript superscript x f apostrophe left parenthesis x right parenthesis space d x space equals space integral subscript negative 3 end subscript superscript x space g left parenthesis x right parenthesis space d x rightwards double arrow f left parenthesis x right parenthesis equals equals space integral subscript negative 3 end subscript superscript x space g left parenthesis x right parenthesis space d x plus f left parenthesis negative 3 right parenthesis space space... space left square bracket S i n c e space f left parenthesis negative 3 right parenthesis space c a n space t a k e space m i n i m u m space v a l u e space a t space minus 1 right square bracket S i n c e comma space g " left parenthesis x right parenthesis greater than 0 rightwards double arrow C u r v e space i s space o p e n i n g space u p w a r d s space f o r space a n y space g left parenthesis x right parenthesis space s a t i s f y i n g space t h e space c o n d i t i o n s C a s e space 1 colon space g left parenthesis x right parenthesis space i s space d e c r e a sin g space integral subscript negative 3 end subscript superscript x space g left parenthesis x right parenthesis space d x greater than 3 cross times 2 square root of 2 equals 6 square root of 2 space space space left parenthesis A r e a space o f space r e c tan g l e right parenthesis C a s e space 2 colon space g left parenthesis x right parenthesis space i s space i n c r e a sin g space integral subscript negative 3 end subscript superscript x space g left parenthesis x right parenthesis space d x greater than 3 cross times 2 square root of 2 equals 6 square root of 2 space space space left parenthesis A r e a space o f space r e c tan g l e right parenthesis C a s e space 1 colon space g left parenthesis x right parenthesis space i s space t a k e space m i n m i m u m space v a l u e space a t space x equals 0 space integral subscript negative 3 end subscript superscript x space g left parenthesis x right parenthesis space d x greater than 3 cross times 2 square root of 2 equals 6 square root of 2 space space space left parenthesis A r e a space o f space r e c tan g l e right parenthesis rightwards double arrow f left parenthesis x right parenthesis greater than 6 square root of 2 minus 1 B u t space f left parenthesis x right parenthesis space c a n apostrophe t space b e space g r e a t e r space t h a n space 1 T h i s space s h o w s space t h e space c o n t r a d i c t i o n space w h i c h space m e a n s space o u r space a s s u m p t i o n space i s space w r o n g rightwards double arrow g left parenthesis x right parenthesis space a n d space g " left parenthesis x right parenthesis space c a n apostrophe t space p o s i t i v e space a t space t h e space s a m e space t i m e S i m i l a r l y comma space t h e y space c a n apostrophe t space b e space n e g a t i v e space a t space t h e space s a m e space t i m e H e n c e comma space t h e r e space e x i s t space c element of left parenthesis negative 3 comma space 3 right parenthesis space s u c h space t h a t space g left parenthesis c right parenthesis g " left parenthesis c right parenthesis less than 0
Answered by Renu Varma | 07 Apr, 2020, 16:06: PM

Application Videos

Concept Videos

Multiple Choice Questions Practice Test
Answered Unanswered My Questions My Answers
JEE main - Maths
in how many ways a team of 2 boys and 4 girls selected from 5 boys and 6 girls
Asked by pallavibirangal87 | 08 Dec, 2024, 10:18: AM
ANSWERED BY EXPERT ANSWERED BY EXPERT
JEE main - Maths
if a chord of length 18 cm is 12 cm away from the centre then find the distance of 24 cm long chord from the centre
Asked by pritisindra6 | 06 Dec, 2024, 21:04: PM
ANSWERED BY EXPERT ANSWERED BY EXPERT
JEE main - Maths
The number of ways in which at least one coin can be selected from 3 coins of one rupee, 4 coins of fifty paise, 5 coins of twenty paise and 6 coins of ten paise is-
Asked by kanishthkumawat | 29 Nov, 2024, 00:23: AM
ANSWERED BY EXPERT ANSWERED BY EXPERT
JEE main - Maths
what is range of tan^-1 x
Asked by pawandhayal125 | 06 Oct, 2024, 10:03: AM
ANSWERED BY EXPERT ANSWERED BY EXPERT
JEE main - Maths
what is 2+2
Asked by srishtipra2008 | 28 Sep, 2024, 10:33: AM
ANSWERED BY EXPERT ANSWERED BY EXPERT
JEE main - Maths
n arithmetic means are inserted between the numbers 7 and 49. if the sum of these means is 364 then the sum of their squares is
Asked by yogeshsoni575 | 23 Sep, 2024, 23:01: PM
ANSWERED BY EXPERT ANSWERED BY EXPERT
JEE main - Maths
consider a dodecagon then number of ways of selecting two triangles which can be made using the vertices of dodecagon so that they do not have a common side.
Asked by sumit.parsudih | 17 Sep, 2024, 22:22: PM
ANSWERED BY EXPERT ANSWERED BY EXPERT
JEE main - Maths
the sum of infinite series 1/2(1/3+1/4)-1/4(1/3^2+1/4^2)+1/6(1/3^3+1/4^3)+
Asked by pritamlovevanshi5 | 15 Sep, 2024, 08:24: AM
ANSWERED BY EXPERT ANSWERED BY EXPERT
JEE main - Maths
solve the equation 6x-y+3z=10 , 3x+6y=18 , x+y+5z=-10 by jacobis method with (3,0,-2) as the initial approcimate to the solution
Asked by ganeshbabugottapu7 | 12 Sep, 2024, 18:19: PM
ANSWERED BY EXPERT ANSWERED BY EXPERT
JEE main - Maths
find the inverse of a Matrix by using gauss Jordan method -2. 1. 3 A = 0. -1. 1 1. 2. 0
Asked by ganeshbabugottapu7 | 10 Sep, 2024, 07:32: AM
ANSWERED BY EXPERT ANSWERED BY EXPERT
My Class Videos Tests ₹ Plans Ask a Doubt