I realize that Lagrange multipliers are extremely useful for applied optimization problems.
However, I know that the standard analytic proof of the spectral theorem relies on them. I've also seen a few other uses/mentions of them in some pure math textbooks. (For example, Wade's An Introduction to Analysis uses an exercise on Lagrange multipliers to later prove a result due to Bernstein on the convergence of Fourier series.)
My question, then, is if Lagrange multipliers are generally a useful technique for extremal problems that arise in pure math. If so, are there some well-known proofs in this area that use them (other than those I mentioned)?
I'm simply curious as to their use outside applied optimization, since the derivation of the existence of the so-called Lagrange multiplier is really just a corollary of a very geometric fact--namely that the gradient is perpendicular to the level sets.
EDIT: To be a bit more specific, by "useful" I mean that it is in fact applicable to certain pure math problems with somewhat regular frequency.