Fundamental lemma of calculus of variations with second derivative

Intense debate at work place around the solution for this:

Let $M \in C[a,b]$ be a continuous function on the closed interval $[a,b]$ that satisfies $$\int_{a}^{b}M(x)\eta^{\prime\prime}dx = 0,$$ for all $\eta \in C^{2}\left[a,b\right]$ satisfying $\eta(a) = \eta(b) = \eta^{\prime}(a) = \eta^{\prime}(b) = 0$. Prove that $M(x) = c_{0}+c_{1}x$ for suitable $c_{0}$ $c_{1}$. What can you say about $c_{0}$, $c_{1}$?

I tried to use integration by parts, and use the fundamental lemma of the calculus of variations, and the lemma of Du Bois and Reynolds to prove it, but that requires $M$ to be $C^1([a, b])$.

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