Question

If a function satisfies $(x-y)f(x+y)-(x+y)f(x-y)=2(x^{2}y-y^3),\forall x,y\in R$ and $f\left(1\right)=2,$ then

• A. $f\left(x\right)$ must be polynomial function
• B. $f\left(3\right)=12$
• C. $f\left(0\right)=0$
• D. $f\left(x\right)$ may not be differentiable

Answer 1

Answer: (A), (B), (C)

The correct options are
A $f\left(x\right)$ must be polynomial function
B $f\left(3\right)=12$
C $f\left(0\right)=0$

$(x-y)f(x+y)-(x+y)f(x-y)$ $=2y\left(\right(x-y\left)\right(x+y\left)\right)$

Let $x-y=u,x+y=v$

$uf\left(v\right)-vf\left(u\right)=uv(v-u)$

$\frac{f\left(v\right)}{v}-\frac{f\left(u\right)}{u}=v-u$

$\Rightarrow (\frac{f\left(v\right)}{v}-v)=(\frac{f\left(u\right)}{u}-u)=constant$

Let $\frac{f\left(x\right)}{x}-x=\lambda$

$\Rightarrow f\left(x\right)=(\lambda x+x^2)$

$f\left(1\right)=2$

$\lambda +1=2\Rightarrow \lambda =1$

$f\left(x\right)=x^2+x$