Lagrange multipliers and constrained optimization

How do you maximize a multivariable function when there's some constraint on the inputs that you're allowed? This is the kind of problem that comes up all the time in practice, so knowing how to approach it is incredibly useful.

Constrained optimization introduction

Lagrange multipliers, using tangency to solve constrained optimization

Finishing the intro lagrange multiplier example

Lagrange multiplier example, part 1

Lagrange multiplier example, part 2

The Lagrangian

Meaning of the Lagrange multiplier

Proof for the meaning of Lagrange multipliers