1. The Problem Reframed: Constrained vs. Free Extrema
A constrained extremum is the largest or smallest value a function $f$ attains not over the whole plane, but only among points that satisfy an extra equation $g(x,y)=c$. That equation is the constraint, and the set of points satisfying it — a curve in the plane, or a surface in space — is the constraint set.
Motivating Picture
Consider the plane $z = 2x + 3y$. As a whole it has no highest or lowest point — it extends to $\pm\infty$. But if we restrict attention to the circle $x^2+y^2=9$, then the plane's height on that circle does have a maximum and a minimum. Those are constrained extrema.
The Substitution Route — Honest Acknowledgement
Your first instinct might be: solve the constraint for one variable and substitute. If $g(x,y)=c$ can be rewritten as $y=h(x)$, then you reduce the problem to a single-variable calculus problem: extremize $\tilde{f}(x)=f(x,h(x))$.
When it works: For simple constraints (e.g., $2x+y=6 \Rightarrow y=6-2x$), substitution is fast and clean.
When it fails or hurts:
- Unsolvable constraints: Many constraints cannot be solved explicitly for a variable. Try solving $x^3 + \sin(xy) = 1$ for $y$ — you can't.