Lagrange Multipliers

Here's part of the explanation of constrained optimization from Hughes-Hallett et al. Applied Calculus 4/e (p. 383):


QUESTION

Why does this method work?


This is an introduction to an algorithm for solving constrained optimization problems.

Before introducing an algorithm, better to introduce some context and concepts:


Here's another explanation from the same book (p. 382).

This makes it clearer why there is a tangency relationship at the optimum. But it still treats the constraint as an equation rather than as a function.