This post is going to be about finding the maxima or minima of a function subject to some constraints. This is usually introduced in a multivariate calculus course, unfortunately (or fortunately?) I never got the chance to take a multivariate calculus course that covered this topic. In my undergraduate class, computer engineers only took three half year engineering calculus courses, and the fourth one (for electrical engineers) seems to have covered other basic multivariate calculus topics such as all the various theorems such as Green's, Gauss', Stokes' (I could be wrong though, I never did take that course!). You know what I always imagined Newton saying, "It's never too late to learn multivariate calculus!".
In that vein, this post will discuss one widely used method for finding optima subject to constraints: Lagrange multipliers. The concepts behind it are actually quite intuitive once we come up with the right analogue in physical reality, so as usual we'll start there. We'll work through some problems and hopefully by the end of this post, this topic won't seem as mysterious anymore .