Let f(x) and g(x) be twice differentiable functions on [0,2] satisfying f''(x)=g''(x),f'(1)=6,f(2)=3 and g(2)=9.Then what is f(x)-g(x) at x=4 equal to?
 
 

Asked by Deebu | 5th Sep, 2016, 03:29: PM

Expert Answer:

begin mathsize 20px style f apostrophe apostrophe left parenthesis x right parenthesis space equals space g apostrophe apostrophe left parenthesis x right parenthesis space space
rightwards double arrow f apostrophe left parenthesis x right parenthesis equals space g apostrophe left parenthesis x right parenthesis plus c.......... left parenthesis E q u. space 1 right parenthesis space space space space left parenthesis I n t e g r a t i o n space o n space b o t h space t h e space s i d e s space 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
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N o w comma space f apostrophe left parenthesis 1 right parenthesis space equals space 6 space
S o space comma 6 space equals space g apostrophe left parenthesis 1 right parenthesis space plus space c space........ left parenthesis b y space left parenthesis 1 right parenthesis space right parenthesis
space rightwards double arrow space 6 space minus space g apostrophe left parenthesis 1 right parenthesis equals space c
S u b s t i t u t i n g space i n space e q u a t i o n space 2 space w e space g e t colon
f left parenthesis x right parenthesis space equals space g left parenthesis x right parenthesis space plus space left parenthesis 6 space minus space g apostrophe left parenthesis 1 right parenthesis right parenthesis x space plus space d
I t space i s space g i v e n space t h a t space f left parenthesis 2 right parenthesis space equals space 3 space a n d space g left parenthesis 2 right parenthesis space equals space 9
S o comma
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space space space 3 space equals space 9 space plus space 2 left parenthesis 6 space minus space g apostrophe left parenthesis 1 right parenthesis right parenthesis space plus space d
space space space 3 space minus space 9 space equals space 12 space minus space 2 g apostrophe left parenthesis 1 right parenthesis space plus space d
space space space minus 6 space equals space 12 space minus space 2 space g apostrophe left parenthesis 1 right parenthesis space plus space d space
space space space minus 6 space minus 12 space equals negative space 2 space g apostrophe left parenthesis 1 right parenthesis space plus space d space
space space space space minus 18 space plus 2 g apostrophe left parenthesis 1 right parenthesis space equals space d
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P u t t i n g space x space equals 4 space i n space t h e space a b o v e space e q u a t i o n space w e space g e t colon
f left parenthesis 4 right parenthesis space equals space g left parenthesis 4 right parenthesis space plus space left parenthesis 6 space minus space g apostrophe left parenthesis 1 right parenthesis right parenthesis 4 space plus space left parenthesis negative 18 space plus space 2 g apostrophe left parenthesis 1 right parenthesis right parenthesis
f left parenthesis 4 right parenthesis space equals space g left parenthesis 4 right parenthesis space equals space 24 space minus 4 g apostrophe left parenthesis 1 right parenthesis minus space 18 space plus space 2 g apostrophe left parenthesis 1 right parenthesis
f left parenthesis 4 right parenthesis space equals space g left parenthesis 4 right parenthesis plus space 6 space minus space 2 g apostrophe left parenthesis 1 right parenthesis
f left parenthesis 4 right parenthesis space minus space g left parenthesis 4 right parenthesis space equals space 6 space minus 2 g apostrophe left parenthesis 1 right parenthesis space

A s space g space i s space t w i c e space d i f f e r e n t i a b l e space i n space t h e space g i v e n space i n t e r v a l space s o space t h e space v a l u e space o f space g apostrophe left parenthesis 1 right parenthesis space e x i s t s. end style

Answered by Tripti Agrawal | 8th Sep, 2016, 11:17: AM

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