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A function f(x) satisfies the functional...

A function `f(x)` satisfies the functional equation `x^2f(x)+f(1-x)=2x-x^4` for all real `x. f(x)` must be

A

`x^(2)`

B

`1-x^(2)`

C

`1+x^(2)`

D

`x^(2)+x+1`

Text Solution

Verified by Experts

The correct Answer is:
B
`x^(2) F(x)+F(1-x)=2x-x^(4) " " `(1)
Replacing x by `1-x`, we get
`(1-x)^(2) F(1-x)+F(x)=2(1-x)-(1-x)^(4) " " `(2)
Eliminating `F(1-x)` from (1) and (2), we get `F(x)=1-x^(2).`
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