Suppose $f\in C^2([a,b])$ and $f(a)=f(b)=0$,use Lagrange interpolation to prove
$$\max_{x\in[a,b]}|f(x)|\leq\frac{(b-a)^2}{8}\max_{x\in[a,b]}|f''(x)|$$
I tried to use the theoretic error to prove this but somehow things does not go very well and I don't see where I should use the condition $f(a)=f(b)=0$. Any hints for this?
Use the fact that the interpolating polynomial of degree $1$ that passes through $(a,f(a))$ and $(b,f(b))$ is the zero polynomial. Then use the remainder formula with $n=1$: $$ R_n(x) = \frac{f^{n+1}(\xi)}{(n+1)!}(x-x_0)\cdots(x-x_n) $$ where $\xi$ lies between the minimum and maximum of $\{x,x_0,\dots,x_n\}$.
You can also use Taylor's series, but I think this is how you were meant to do the problem.