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 on $\closedint a b$.

Let:


 * $\forall \map h x \in C^1: \ds \int_a^b \paren {\map \alpha x \map h x + \map \beta x \map {h'} x} \rd x = 0$

subject to the boundary conditions:
 * $\map h a = \map h b = 0$

Then $\map \beta x$ is differentiable.

Furthermore:


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

Proof
Using Integration by Parts allows us to factor out $\map h x$:

Hence the problem has been reduced to:


 * $\ds \int_a^b \paren {\map \alpha x - \map {\beta'} x} \map h x \rd x = 0$

Since If Definite Integral of $\map a x \map h x$ vanishes for any $C^0 \map h x$ then $C^0 \map a x$ vanishes, the conclusion is that in the considered interval $\closedint a b$ it holds that:


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