So many values are already given

Let $f(x)$ be a non-constant thrice differentiable function defined on real numbers such that $f(x)=f(6-x)$ and $f'(0)=0=f'(2)=f'(5)$ . Find the minimum number of values of $p \in [0,6]$ which satisfy the equation $(f''(p))^2+f'(p)f'''(p)=0$ Details and Assumptions:

• $f'(p)=\left( \frac{df(x)}{dx} \right)_{x=p}$

• $f''(p)=\left( \frac{d^2f(x)}{dx^2} \right)_{x=p}$

• $f'''(p)=\left( \frac{d^3f(x)}{dx^3} \right)_{x=p}$