2. Take a moment to understand the Lagrange multipliers method for finding min/max on a surface f(x, y, z) subject to a constraint g(x,y, z) = 0. Geometrically speaking, what does it mean for Vf = AVg? Why are the candidates for minimums & maximums exactly the points that satisfy this equation? %3D

Suppose we want to find the maximum of a function $f\left(x,y\right)=xy$ subject to a constraint $g\left(x,y\right)=\frac{x^2}{9}+\frac{y^2}{4}-1$.

To see how Lagrange multipliers work, look at the following graph:

Figure 1

The graph shows the function f from above along with constraint g = c and some level curves of f. The constraint is nothing but a plane that cuts through the function f.

The constraint g is an ellipse on xy-plane projected on to the surface f. The goal here is to find the maximum value of f without moving outside the elliptic boundary.