In the proof of the Euler-Lagrange equation, the final step invokes a lemma known as the fundamental lemma of the calculus of variations (FLCV).
Here is a sketch of the proof. Suppose, for a contradiction, that for some a < α < b we have y(α) > 0 (the case when α = a or α = b can be done similarly, but let's keep it simple). Because y is continuous, y(x) > 0 for all x in some interval (α0, α1) containing α.
Consider the function η : [a, b]→ defined by
η(x) = |
By hypothesis, y(x)η(x) dx = 0. But y(x)η(x) is continuous, zero outside (α0, α1), and strictly positive for all x∈(α0, α1). A strictly positive continuous function on an interval like this has a strictly positive integral, so this is a contradiction. Similarly we can show y(x) never takes values < 0, so it is zero everywhere on [a, b].