
We see how to find extrema of functions of several variables.

Given a function $z=F(x,y)$, we are often interested in points where $z$ takes on the largest or smallest values. For instance, if $z$ represents a cost function, we would likely want to know what $(x,y)$ values minimize the cost. If $z$ represents the ratio of a volume to surface area, we would likely want to know where $z$ is greatest. This leads to the following definition that we will state rather generally, but use mostly in the case of a function $F:\R ^2\to \R$.

Critical points

If $F$ has a local or absolute maximum at $\vec {c}$, it means the gradient will point “nowhere” since the gradient points in the initial direction of greatest increase. This means it is pointing in a “direction” whose components are either zero or undefined. In an entirely similar way, the gradient will be a vector whose components are either zero or undefined at local minimums as well.

Therefore, to find local extrema, we find the critical points of $F$ and determine which correspond to local maxima, local minima, or neither. We’ll use examples to demonstrate this process.

In each of the previous two examples, we found a critical point of $F$ and then determined whether or not it was a local (or absolute) maximum or minimum by graphing. It would be nice to be able to determine whether a critical point corresponded to a max or a min without a graph. Before we develop such a test, we do one more example that sheds more light on the issues our test needs to consider.

The most obvious example of a saddle point is a the point determined by $\vector {0,0}$ on a hyperbolic paraboloid of the form $z = \pm x^2 \mp y^2$.

When thinking about a graph of $z= F(x,y)$ at a saddle point, the instantaneous rate of change in all directions is $0$ and there are points nearby with $z$-values both less than and greater than the $z$-value of the saddle point.

The second derivative test

In theory to identify local extrema verses saddle points, we could compute the Taylor polynomial of degree $2$ at the critical point in question, and then identify the Taylor polynomial as either:

Elliptic paraboloid
Indicating we have found local extrema.
Hyperbolic paraboloid
Indicating that we are at a saddle point.

Fortunately, as we have seen, there is a second derivative test that does exactly this for us. We will now restate this test in the context of identifying local extrema.

We first practice using this test with the function in the previous example, where we visually determined we had a local maximum and a saddle point.

For some interesting extra reading check out: