We see how to find extrema of functions of several variables.
- If there is an open disk containing such that for all in , then has a local maximum at ; if for all in , then has a local minimum at .
- If for all in , then has an absolute maximum at ; if for all in , then has an absolute minimum at .
- If has a local maximum or minimum at , then has a local extrema at ; if has an absolute maximum or minimum at , then has a absolute extrema at .
If has a local or absolute maximum at , 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 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 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.
We have two critical points: and . To determine if they correspond to a local maximum or minimum, we consider the graph of below:
The critical point clearly corresponds to a local maximum. However, the critical point at is neither a maximum nor a minimum, displaying a different, interesting characteristic.
If one walks parallel to the -axis towards this critical point, then this point becomes a local maximum along this path. But if one walks towards this point parallel to the -axis, this point becomes a local minimum along this path. A point that seems to act as both a max and a min is a saddle point. A formal definition follows.
The most obvious example of a saddle point is a the point determined by on a hyperbolic paraboloid of the form .
When thinking about a graph of at a saddle point, the instantaneous rate of change in all directions is and there are points nearby with -values both less than and greater than the -value of the saddle point.
In theory to identify local extrema verses saddle points, we could compute the Taylor polynomial of degree 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.
- If and then is a local maximum.
- If and then is a local minimum.
- If , then is a saddle point.
- If , the test is inconclusive.
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.
The second derivative test has confirmed the visual evidence we found before.
We find the critical points by finding where , since the partial derivatives are defined everywhere on . With a bit of algebra we can show that there are three critical points
- , and
Now for each critical point , we compute compute . We find
- Since , we see is also a saddle point.
- Since , we see is a saddle point.
- Since and , we see is a local maximumminimum .
Below we see a graph of and the three critical points.
For some interesting extra reading check out:
- Three Observations on a Theme: Editorial Note, D.A. Smith, Mathematics Magazine, May 1985.
- A Surface with One Local Minimum, J.M. Ash and H. Sexton, Mathematics Magazine, May 1985.
- “The Only Critical Point in Town” Test, I. Rosenholtz and L. Smylie, Mathematics Magazine, May 1985.
- Two Mountains Without a Valley, I. Rosenholtz, Mathematics Magazine, February 1987.