$\text{ Let }f:R\to R\text{ is a polynomial function which satisfies }$$f(x+y)+f(x-y)=2f\left(x\right)-2y^2\forall x,y\in R,f^{\text{'}}\left(0\right)\ne 0\text{ and }f\left(0\right)=0\text{ . Then which of }$the following is/are true 

$\begin{array}{l}put x=0 in the given equation then f\left(y\right)+f(-y)=-2y^2 \left(\sin ce f\left(0\right)=0\right)\text{ and}\\ put y=x in the given equation then f\left(2x\right)-2f\left(x\right)=-2x^2\\ f\left(x\right)\text{ must be quadratic }\\ f\left(x\right)=-x^2+\lambda x\text{ where }\lambda =f^1\left(0\right)\\ \text{ area }={\int }_0^{\lambda } \left(-x^2+\lambda x\right)dx=\frac{{\lambda }^3}{6}\end{array}$