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 \left [ { \alpha \left ( { x } \right ) h \left ( { x } \right ) + \beta \left ( { x } \right ) h' \left ( { x } \right) } \right ] \mathrm d x = 0 \quad \forall h \left ( { x } \right ) \in C^1 : h \left ( { a } \right ) = h \left ( { b } \right ) = 0 $,

Then $\map \beta x$ is differentiable.

Furthermore:


 * $ \beta' \left ( { x } \right ) = \alpha \left ( { x } \right ) \quad \forall x \in \closedint a b$.

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

Hence, the problem has been reduced to


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

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= \beta' \left ( { x } \right ) $