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Let f be a differential function such th...

Let `f` be a differential function such that `f(x)=f(2-x)` and `g(x)=f(1+x)` then

A

g(x) is an odd function

B

g(x) is an even function

C

Graph of `f(x)` is symmetrical about the line `x=1`

D

`f'(1)=0`

Text Solution

Verified by Experts

The correct Answer is:
B, C, D
We have `f(x)=f(2-x)`
Replacing x by `f(1+x)`
`f(+x)=f(1-x)`
Hence graph of `f(x)` is symmetrical about the line `x=1`
Also `g(x)=f(1+x)=f(1-x)=g(-x)`
` :. g(x) ` is an even function.
Further `f'(1+x)= -f'(1-x)`
` :. f'(1)= -f'(1)`
` :. f'(1)=0`
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