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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 6th April 2020, 11:40 PM
Answered by 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 Expert 7th April 2020, 4:06 PM
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