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If a function satisfies (x-y)f(x+y)-(x+y...

If a function satisfies `(x-y)f(x+y)-(x+y)f(x-y)=2(x^2 y-y^3) AA x, y in R and f(1)=2`, then

A

f(x) must be polynomial function

B

f(3) = 12

C

f(0) = 0

D

f(x) may not be differentiable

Text Solution

Verified by Experts

The correct Answer is:
A, B, C
`(x-y)f(x+y)-(x+y)f(x-y)=2y(x-y)(x+y)`
Let `x-y=u, x+y=v`. Then
`uf(v)-vf(u)=uv(v-u)`
`rArr" "(f(v))/(v)-(f(u))/(u)=v-u`
`rArr" "((f(v))/(x)-v)=((f(u))/(u)-u)=" constant"`
`"Let "(f(x))/(x)-x=lambda. " Then"`
`f(x)=(lambdax+x^(2))`
Since `f(1)=2" then "lambda=1`
`therefore" "f(x)=x^(2)+x`
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