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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 (1) g(x) is an odd function (2) g(x) is an even function (3) graph of f(x) is symmetrical about the line x= 1 (4) `f'(1)=0`

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)`, we get
`f(1+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)`
`therefore` g(x) is an even function.
Further `f'(1+x)=-f'(1-x)`
`therefore" "f'(1)=-f'(1)`
`therefore" "f'(1)=0`
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