Prove that $\mathop{max}_{x\in [0,1]}|u(x)|\le \frac{1}{8}\mathop{max}_{x\in [0,1]}|{u''(x)}|$ when $u(x)\in C^2[0,1],u(0)=u(1)=0$

Question

I want to prove the inequality with analysis methods instead of method of PDE

Actually we can consider the following PDE: $$\left\{\begin{array}{l} u''(x)=u''(x)\\ u(0)=u(1)=0 \end{array}\right.$$ and we can get $u(y)$ by Green's function directly: G(x; y), that is $u(y)=\int_{0}^{1}u''(x)G(x;y)dx$

estimate the formula and we get the solution.

But I want to prove the inequality with analysis methods, are there any hints?