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


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Question Number 10039 by Gaurav3651 last updated on 21/Jan/17

$l e t f (x) a n d g (x) b e t w i c e d i f f e r e n t i a b l e$ $f u n c t i o n s o n [0, 2] s a t i s f y i n g$ $f^{″} (x) = g^{″} (x), f^{'} (1) = 4, g^{'} (1) = 6,$ $f (2) = 3 a n d g (2) = 9 . T h e n w h a t i s$ $f (x) - g (x) a t x = 4 e q u a l t o ?$
Commented by prakash jain last updated on 22/Jan/17

$f^{″} (x) = g^{″} (x)$ $\Rightarrow f^{'} (x) = g^{'} (x) + C$ $f^{'} (1) = 4, g^{'} (1) = 6$ $\Rightarrow 4 = 6 + C \Rightarrow C = - 2$ $f^{'} (x) = g^{'} (x) - 2$ $i n t e g r a t i n g$ $f (x) = g (x) - 2 x + C_{1}$ $f (2) = 3, g (2) = 9$ $3 = 9 - 2 \cdot 2 + C_{1} \Rightarrow C_{1} = - 2$ $f (x) = g (x) - 2 x - 2$ $f (x) - g (x) = - 2 x - 2$ $f (4) - g (4) = - 2 \cdot 4 - 2 = - 10$
Commented by mrW1 last updated on 22/Jan/17

$GREAT$
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