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f:[0,5]rarrR,y=f(x) such that f''(x)=f''...

`f:[0,5]rarrR,y=f(x)` such that `f''(x)=f''(5-x)AAx in [0,5] f'(0)=1 and f'(5)=7`, then the value of `int_(1)^(4)f'(x)dx` is

A

4

B

6

C

8

D

10

Text Solution

Verified by Experts

The correct Answer is:
C
`int_(1)^(4)f'(x)dx=[xf'(x)]_(1)^(4)-int_(1)^(4)xf''(x)`
Now `I=int_(1)^(4)xf''(x)dx=int_(1)^(4)(5-x)f''(5-x)dx`
`=5int_(1)^(4)f''(x)dx=-I`
`therefore" "I=(5)/(2)[f'(4)-f'(1)]`
`therefore" "int_(1)^(4)f'(x)dx=(3)/(2)[f'(4)+f'(1)]`
Now, `f''(x)=f''(5-x)`
`rArr" "f'(x)=-f'(5-x)+c`
`rArr" "f'(0)+f'(5)=c rArr c=8`
`"so "f'(x)+f'(5-x)=8 rArr f'(4)+f'(1)=8`
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