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

Theorem
Let $\map \alpha x,\map \beta x$ be real functions.

Let $\map \alpha x,\map \beta x$ be continuous in $\closedint a b$.

Let:


 * $\displaystyle\int_a^b \sqbrk{\map \alpha x \map h x+\map \beta x \map {h'} x} \d x = 0 \quad \forall \map h x\in C^1 :\map h a=\map h b=0$,

Then $\map \beta x$ is differentiable.

Furthermore:


 * $ \map {\beta'} x=\map \alpha x\quad\forall x\in\closedint a b$.

Proof
Using integration by parts allows us to factor out $\map h x$:

Hence, the problem has been reduced to


 * $\displaystyle \int_a^b \sqbrk{ \map \alpha x-\map {\beta'} x}\map h x \d x=0$

Since 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 $\closedint a b$ it holds that


 * $\map \alpha x=\map {\beta'} x$