Necessary Condition for Integral Functional to have Extremum for given function/Lemma

Theorem
If $\alpha\left(x\right)$ and $\beta\left(x\right)$ are continuous in $\left[{{a}\,.\,.\,{b}}\right]$ and if $\displaystyle\int_{a}^{b}\left[\alpha\left(x\right)h\left(x\right)+\beta\left(x\right)h'\left(x\right)\right]\mathrm{d}{x}=0$ for every function $h\left(x\right)\in \mathscr{D}_1$ such that $h\left(a\right)=0$ and $h\left(b\right)=0$, then $\beta\left(x\right)$ is differentiable, and $\beta'\left(x\right)=\alpha\left(x\right)$ for all $x$ in $\left[{{a}\,.\,.\,{b}}\right]$.

Proof
Using integration by parts allows us to factor out $h\left(x\right)$:

Hence, the problem has been reduced to

$\displaystyle \displaystyle\int_{a}^{b}\left[\alpha(x)-\beta'(x)\right]h(x)\mathrm{d}{x}=0$

By If Definite Integral of a(x)h(x) vanishes for any C^0 h(x) then C^0 a(x) vanishes, the conclusion is that in the considered interval $\left[{{a}\,.\,.\,{b}}\right]$ it holds that

$\alpha(x)=\beta'(x)$