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Section 11.1 Optimization

Before considering functions of several variables, let us first review how to find maxima and minima for functions of one variable. Recall that a local max/min can only occur at a critical point, where the derivative either vanishes or is undefined. The second derivative test can help to determine whether a critical point is a max or a min: If the second derivative is positive or negative, then the graph is concave up or down, respectively, and the critical point is a local min or a local max, respectively. However, if the second derivative vanishes, anything can happen.

Now imagine the graph of a function of two variables. At a local minimum, you would expect the slope to be zero, and the graph to be concave up in all directions. A typical example would be

\begin{equation*} f(x,y) = x^2 + y^2 \end{equation*}

at the origin, as shown in Figure 11.2.1. Similarly, at a local maximum, the slope would again be zero but the graph would be concave down in all directions; a typical example would be

\begin{equation*} g(x,y) = -x^2 - y^2 \end{equation*}

again at the origin, as shown in Figure 11.2.2. But now there is another possibility. Consider the function

\begin{equation*} h(x,y) = x^2 - y^2 \end{equation*}

whose graph still has slope zero at the origin, but which is concave up in the \(x\)-direction, yet concave down in the \(y\)-direction; this is a saddle point, as shown in Figure 11.2.3.

Motivated by the above examples, a critical point of a function \(f\) of several variables occurs where all of the partial derivatives of \(f\) either vanish or are undefined.

Critical points of a function \(f(x,y)\) of two variables can be classified using the second derivative test, which now takes the following form. Let

\begin{align*} D \amp= \frac{\partial^2 f}{\partial x^2}\frac{\partial^2 f}{\partial y^2} - \left(\frac{\partial^2 f}{\partial x\partial y}\right)^2\\ A \amp= \frac{\partial^2 f}{\partial x^2} \end{align*}

evaluated at a critical point \(P=(a,b)\text{.}\) Then

  • If \(D>0\) and \(A>0\text{,}\) then \(f(a,b)\) is a local min.

  • If \(D>0\) and \(A\lt 0\text{,}\) then \(f(a,b)\) is a local max.

  • If \(D\lt 0\text{,}\) then there is a saddle point at \(P\text{.}\)

  • If \(D=0\text{,}\) anything can happen.

Optimization problems typically seek a global max/min, rather than a local max/min. Just as for functions of one variable, in addition to finding the critical points, one must also examine the boundary. Thus, to optimize a function \(f\) of several variables, one must:

  • Find the critical points.

  • Find any critical points of the function restricted to the boundary.

  • Evaluate \(f\) at each of these points to find the global max/min.

The middle step is effectively a single-variable optimization problem, which may require considering different pieces of the boundary separately, together with any “corner points”.

As an example, consider the function \(f(x,y)=xy\text{.}\) Where are the critical points? Where the partial derivatives of \(f\) vanish. We have

\begin{equation} \Partial{f}{x} = y ;\qquad \Partial{f}{y} = x\tag{11.1.1} \end{equation}

so the only critical point occurs where \(x=0=y\text{,}\) that is, at the origin. We compute second derivatives and evaluate them at the origin, obtaining

\begin{equation} \frac{\partial^2 f}{\partial x^2}\Bigg|_{(0,0)} = 0 = \frac{\partial^2 f}{\partial y^2}\Bigg|_{(0,0)} ; \qquad \frac{\partial^2 f}{\partial x\partial y}\Bigg|_{(0,0)} = 1\tag{11.1.2} \end{equation}

so that in this case

\begin{equation} D = 0 - 1 = -1 \lt 0\tag{11.1.3} \end{equation}

which implies that \((0,0)\) is a saddle point of \(f\text{.}\)

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