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

Theorem
Let $ \alpha \left ( { x } \right ) $, $ \beta \left ( { x } \right ) $ be real functions.

Let $ \alpha \left ( { x } \right ) $, $ \beta \left ( { x } \right ) $ be continuous in $ \left [ { a \,. \,. \, b } \right ] $.

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 $ \beta \left ( { x } \right ) $ is differentiable.

Furthermore:


 * $ \beta' \left ( { x } \right ) = \alpha \left ( { x } \right ) \quad \forall 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 \left ( { x } \right )-\beta' \left ( { x } \right ) } \right ] h \left ( { x } \right ) \mathrm 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 $ \left [ {a \,. \,. \, b } \right ] $ it holds that


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