Let $\frac{f\left(x+y\right)-f\left(x\right)}{2}=\frac{f\left(y\right)-a}{2}+xy$ for all real $xandy\dot{}$ If $f\left(x\right)$ is differentiable and $f^{prime}\left(0\right)$ exists for all real permissible value of $a$ and is equal to $\sqrt{5a-1-a^2}\dot{}$ Then $f\left(x\right)$ is positive for all real $x$$f\left(x\right)$ is negative for all real $x$$f\left(x\right)=0$ has real roots Nothing can be said about the sign of $f\left(x\right)$

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