APEX Calculus

Given a function \(f(x,y)\text{,}\) we are often interested in points where \(z=f(x,y)\) takes on the largest or smallest values. For instance, if \(f\) represents a cost function, we would likely want to know what \((x,y)\) values minimize the cost. If \(f\) represents the ratio of a volume to surface area, we would likely want to know where \(f\) is greatest. This leads to the following definition.

If \(f\) has a relative or absolute maximum at \((x_0,y_0)\text{,}\) it means every curve on the graph of \(f\) through \((x_0,y_0,f(x_0,y_0))\) will also have a relative or absolute maximum at \(P\text{.}\) Recalling what we learned in Section 3.1, the slopes of the tangent lines to these curves at \(P\) must be 0 or undefined. Since directional derivatives are computed using \(f_x\) and \(f_y\text{,}\) we are led to the following definition and theorem.

Therefore, to find relative extrema, we find the critical points of \(f\) and determine which correspond to relative maxima, relative minima, or neither. The following examples 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 relative (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.

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.

Before Example 13.8.8 we mentioned the need for a test to differentiate between relative maxima and minima. We now recognize that our test also needs to account for saddle points. To do so, we consider the second partial derivatives of \(f\text{.}\)

Recall that with single variable functions, such as \(y=f(x)\text{,}\) if \(\fp'(c) \gt 0\text{,}\) then \(f\) is concave up at \(c\text{,}\) and if \(\fp(c) =0\text{,}\) then \(f\) has a relative minimum at \(x=c\text{.}\) (We called this the Second Derivative Test.) Note that at a saddle point, it seems the graph is “both” concave up and concave down, depending on which direction you are considering.

It would be nice if the following were true:

However, this is not the case. Functions \(f\) exist where \(f_{xx}\) and \(f_{yy}\) are both positive but a saddle point still exists. In such a case, while the concavity in the \(x\)-direction is up (i.e., \(f_{xx} \gt 0\)) and the concavity in the \(y\)-direction is also up (i.e., \(f_{yy} \gt 0\)), the concavity switches somewhere in between the \(x\)- and \(y\)-directions.

To account for this, consider \(D = f_{xx}f_{yy}-f_{xy}f_{yx}\text{.}\) Since \(f_{xy}\) and \(f_{yx}\) are equal when continuous (refer back to Theorem 13.3.11), we can rewrite this as \(D = f_{xx}f_{yy}-f_{xy}^{\,2}\text{.}\) \(D\) can be used to test whether the concavity at a point changes depending on direction. If \(D \gt 0\text{,}\) the concavity does not switch (i.e., at that point, the graph is concave up or down in all directions). If \(D\lt 0\text{,}\) the concavity does switch. If \(D=0\text{,}\) our test fails to determine whether concavity switches or not. We state the use of \(D\) in the following theorem.

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

When optimizing functions of one variable such as \(y=f(x)\text{,}\) we made use of Theorem 3.1.3, the Extreme Value Theorem, that said that over a closed interval \(I=[a,b]\text{,}\) a continuous function has both a maximum and minimum value. To find these maximum and minimum values, we evaluated \(f\) at all critical points in the interval, as well as at the endpoints (the “boundary”) of the interval.

A similar theorem and procedure applies to functions of two variables. A continuous function over a closed set also attains a maximum and minimum value (see the following theorem). We can find these values by evaluating the function at the critical values in the set and over the boundary of the set. After formally stating this extreme value theorem, we give examples.

This portion of the text is entitled “Constrained Optimization” because we want to optimize a function (i.e., find its maximum and/or minimum values) subject to a constraint — some limit to what values the function can attain. In the previous example, we constrained ourselves by considering a function only within the boundary of a triangle. This was largely arbitrary; the function and the boundary were chosen just as an example, with no real “meaning” behind the function or the chosen constraint.

However, solving constrained optimization problems is a very important topic in applied mathematics. The techniques developed here are the basis for solving larger problems, where more than two variables are involved.

We illustrate the technique once more with a classic problem.

It is hard to overemphasize the importance of optimization. In “the real world,” we routinely seek to make something better. By expressing the something as a mathematical function, “making something better” means “optimize some function.”

The techniques shown here are only the beginning of an incredibly important field. Many functions that we seek to optimize are incredibly complex, making the step of “find the gradient and set it equal to \(\vec 0\)” highly nontrivial. Mastery of the principles here are key to being able to tackle these more complicated problems.