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Given f'(1)=1and f(2x)=f(x)AAxgt0.If f'...

Given `f'(1)=1and f(2x)=f(x)AAxgt0.If f'(x)` is differentiable, then there exists a number `c in (2,4)` such that `f''(c )` equal

A

`1//4`

B

`-1//2`

C

`-1//4`

D

`-1//8`

Text Solution

Verified by Experts

The correct Answer is:
D
Given `f'(1)=1,`
And `f(2x)=f(x)`
`rArr" "2f'(2x)=f'(x)`
Putting x = 1, `f'(2)=(f'(1))/(2)=(1)/(2)`
Putting x = 2, `f'(4)=(f'(2))/(2)=(1)/(4)`
Now applying LMVT for `y=f'(x)` in `[2, 4]`
We get `f''(c)=(f'(4)-f'(2))/(2)=((1)/(4)-(1)/(2))/(2)=-(1)/(8)`
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