(Theory behind …) When … is solved for …, it gives a…

These problems develop the theory of Lagrange multipliers.

(Theory behind $\lambda$) When $g(x,y)=k$ is solved for $y$, it gives a curve $y=R(x)$. Then minimizing $f(x,y)$ along this curve yields
$$\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}\frac{dR}{dx}=0,\frac{\partial g}{\partial x}+\frac{\partial g}{\partial y}\frac{dR}{dx}=0.$$
Those come from the ______ rule: $df/dx=0$ at the minimum and $dg/dx=0$ along the curve because $g=$______. Multiplying the second equation by $\lambda =(\partial f/\partial y)/(\partial g/\partial y)$ and subtracting from the first gives ______. Also $\partial f/\partial y=\lambda \partial g/\partial y$. These are the equations (1) for $x,y,\lambda$.

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §13.7 Constraints and Lagrange Multipliers, #16