Absolute Maxima and Minima

Functions of One Variable

In Calc I you have learned about absolute maxima and minima of function. Two principles were important for finding those:

A continuous function on a closed interval has both absolute maximum and absolute minimum.
An absolute extrema (a maximum or minimum) on a closed interval can be attained either at a critical point inside the interval, or at an end point of the interval.

To find the absolute maximum or minimum of a continuous function on a closed interval, one has to find all the critical points inside the interval, and plug all those points and the endpoints or the interval to the function, to see at which of the points the function will have the highest or the lowest value.

Functions of several variables

The same principles apply in several variables:

Given a set \(A\) of points in \(\mathbb{R}^n\):

A point \(p \in \mathbb{R}^n\) is a boundary point of a set \(A \subseteq \mathbb{R}^n\) if any ball centered at \(p\) contains points that are in \(A\) as well as points that are not in \(A\). A point that belongs to \(A\) but is not a boundary point is called an interior point of \(A\). A point \(p \in A\) is an interior point of \(A\) if there exists a ball \(B\) centered at \(p\) such that \(B \subseteq A\).
A set \(A \subseteq \mathbb{R}^n\) is closed if it contains all of its boundary points.
A set \(A \subseteq \mathbb{R}^n\) is open if it contains none of its boundary points.

Examples

Take for example the sets \(G = \left\{(x,y) \in \mathbb{R}^n \mid x^2 + y^2 < 1\right\}\) and \(F = \left\{(x,y) \in \mathbb{R}^n \mid x^2 + y^2 \le 1\right\}\). The boundary of each of the set is the unit circle \(\left\{(x,y) \in \mathbb{R}^2 \mid x^2 + y^2 = 1\right\}\). The set \(F\) contains all the boundary points, so it is closed, while the set \(G\) does not contain any of the boundary points, so it is open.

The two principles

A continuous function on a bounded closed set in \(\mathbb{R}^n\) has both absolute maximum and absolute minimum.
An absolute extrema (a maximum or minimum) of a continuous function on a bounded closed set in \(\mathbb{R}^n\) can be attained either at a critical point that is an interior point of the set, or at a boundary point of the set.

The procedure

Given a continuous function \(f:\mathbb{R}^n \to \mathbb{R}\) defined on a bounded closed set \(A \subseteq \mathbb{R}^n\), the task is to find points in \(A\) where \(f\) has absolute maximum or absolute minimum. We do it by first finding all the candidate points where an absolute maximum or minimum are possible, then plugging all those points into the function to see which of them will give the highest or lowest value.

The set of candidate points will consist of

Critical points in the interior of the set
Candidate points on the boundary.

Finding candidate points on the boundary can generally be very hard, but if the set \(A\) is nice and the boundary consists of one or more parts that can be parametrized using \(n-1\) variables, then it will be reduced to \(n-1\) dimensional version of the same problem. The following video shows an example for \(n=3\), with the set \(A\) being a cube. The example is very long, because the boundary of the three-dimensional cube consists of 6 squares, and each square in turn has 4 boundary segments. Altogether, critical points must be found inside one cube, six squares, and twelve intervals, and the intervals have eight endpoints among them. You may want to watch it with higher speed, or fast forward through some of the calculations.

The following illustration shows the cube with all the candidate points for maxima or minima: the critical point inside the cube (red), the six critical points inside the sides (green), the twelve critical points on the edges (magenta), and the eight corners (blue):

Simpler examples

The following two examples show finding absolute maxima and minima of two variable functions.

The graph of the function looks like this: