Question

Let $f\left(x\right)$ be defined $\forall x\in R$ and continuous.
Let $f(x+y)-f(x-y)=4xy\forall x,y\in R$ and $f\left(0\right)=0$, then $f\left(x\right)<2x+3$ will have solution :

• A. $(-1,3)$
• B. $(1-\sqrt{5},1+\sqrt{5})$
• C. $(1-\sqrt{5},-1)\cup (3,1+\sqrt{5})$
• D. $(3,1+\sqrt{5})$

Answer 1

The correct option is A $(-1,3)$

Put $x=y$ in the given equation, we get

$f\left(2x\right)-f\left(0\right)=4x^2\Rightarrow f\left(2x\right)=4x^2\Rightarrow f\left(x\right)=x^2$

$f\left(x\right)<2x+3\Rightarrow x^2<2x+3\Rightarrow x^2-2x-3<0$
$\Rightarrow (x-3)(x+1)<0\Rightarrow x\in (-1,3)$