Nathan Wakefield, Christine Kelley, Marla Williams, Michelle Haver, Lawrence Seminario-Romero, Robert Huben, Aurora Marks, Stephanie Prahl, Based upon Active Calculus by Matthew Boelkins

Section3.2Global Optimization

Motivating Questions

What are the differences between finding relative extreme values and global extreme values of a function?

How is the process of finding the global maximum or minimum of a function over the function's entire domain different from determining the global maximum or minimum on a restricted domain?

For a function that is guaranteed to have both a global maximum and global minimum on a closed, bounded interval, what are the possible points at which these extreme values occur?

We have seen that we can use the first derivative of a function to determine where the function is increasing or decreasing, and the second derivative to know where the function is concave up or concave down. This information helps us determine the overall shape and behavior of the graph as well as whether the function has relative extrema.

Remember the difference between a relative maximum and a global maximum: there is a relative maximum of \(f\) at \(x = p\) if \(f(p) \ge f(x)\) for all \(x\) near \(p\text{,}\) while there is a global maximum at \(p\) if \(f(p) \ge f(x)\) for all \(x\) in the domain of \(f\text{.}\)

For instance, in Figure3.18 at the right, we see a function \(f\) that has a global maximum at \(x = c\) and a relative maximum at \(x = a\text{,}\) since \(f(c)\) is greater than or equal to \(f(x)\) for every value of \(x\text{,}\) while \(f(a)\) is only greater than or equal to the value of \(f(x)\) for \(x\) near \(a\text{.}\) Since the function appears to decrease without bound, \(f\) has no global minimum, although there is a relative minimum at \(x = b\text{.}\)

Figure3.18A function \(f\) with a global maximum, but no global minimum.

Our emphasis in this section is on finding the global extreme values of a function (if they exist), either over its entire domain or on some restricted portion.

Example3.19

Let \(f(x) = 2 + \frac{3}{1+(x+1)^2}\text{.}\)

Determine all of the critical numbers of \(f\text{.}\)

Construct a first derivative sign chart for \(f\) and thus determine all intervals on which \(f\) is increasing or decreasing.

Does \(f\) have a global maximum? If so, why, and what is its value and where is the maximum attained? If not, explain why.

Determine \(\lim_{x \to \infty} f(x)\) and \(\lim_{x \to -\infty} f(x)\text{.}\)

Explain why \(f(x) \gt 2\) for every value of \(x\text{.}\)

Does \(f\) have a global minimum? If so, why, and what is its value and where is the minimum attained? If not, explain why.

What is true about \(f'(c)\) when \(c\) is a critical number of \(f\text{?}\)

Remember that on a given interval of the first derivative sign chart for \(f\text{,}\) the derivative \(f'\) should have the same sign everywhere, so you only need to test at a single point on the interval.

What does the chart from (b) tell you? Thinking about the algebraic structure of \(f(x)\) can also support your reasoning.

What happens to \(1+(x+1)^2\) for large values of \(x\text{?}\) What happens to its reciprocal?

\(1+(x+1)^2\gt0\) for every value of \(x\text{.}\)

Think about the long-run behavior of \(f\) that you found in (d).

The denominator of \(f'(x)\) is always at least \(1\) for the same reason that the denominator of \(f(x)\) is always at least \(1\text{,}\) so there are no points \(x=c\) for which \(f'(c)\) does not exist. Thus the only critical number(s) of \(f\) will occur when \(f'(c)=0\text{.}\) This happens when

Thus the only critical number of \(f\) is \(c=-1\text{,}\) with \(f'(-1)=0\text{.}\)

As stated in (a), the denominator of \(f'(x)\) is always at least \(1\text{,}\) so in particular, it is always positive. On our first derivative sign chart, the only critical number we have is \(c=-1\text{,}\) which splits the domain of \(f\) and \(f'\) into the intervals \((-\infty,-1)\) and \((-1,\infty)\text{.}\) Thinking of the factors of \(f'(x)\) as \(-6\text{,}\) \((x+1)\text{,}\) and \(\frac{1}{\left(1+(x+1)^2\right)^2}\text{,}\) the signs of these at \(x=-2\) are \(--+\), with the final product positive. Likewise, the signs of these factors at \(x=0\) are \(-++\), making the final product negative. Therefore \(f'(x)\gt0\) on the interval \(x\lt-1\text{,}\) and \(f'(x)\lt0\) on the interval \(x\gt-1\text{.}\) Hence \(f\) is increasing on \((-\infty,-1)\) and decreasing on \((-1,\infty)\text{.}\)

According to the first derivative test, the point \((-1,f(-1))\) is a relative maximum of \(f\text{.}\) Since this is the only critical point of the function and \(f\) is continuous everywhere, there is nowhere else that \(f\) can change direction. It follows that this point is actually a global maximum.

From an algebraic standpoint, we have already observed that \(1+(x+1)^2\ge1\) for every value of \(x\text{.}\) Consequently,

for every value of \(x\text{.}\) Since \(f(-1)=5\text{,}\) we conclude that the global maximum of \(f\) is \(5\) at \(x=-1\text{.}\)

The symmetry of \(f\) about \(x=-1\) allows us to compute both limits simultaneously. We first note that \(\lim_{x\to\pm\infty}\left(1+(x+1)^2\right)=\infty\) and \(\lim_{x\to\pm\infty}\frac{1}{1+(x+1)^2}=0\text{.}\) Thus

Since the quotient in the formula for \(f(x)\) is always strictly greater than zero, adding \(2\) to the quotient to obtain \(f(x)\) will always yield a value strictly greater than \(2\text{.}\)

Since \(f(x)\gt2\) for every value of \(x\text{,}\) a global minimum of \(f\) (if it existed) would necessarily be larger than \(2\text{.}\) But we showed in (d) that \(\lim_{x\to\pm\infty}f(x)=2\text{,}\) meaning that \(f\) gets arbitrarily close to \(2\) as \(x\) gets large. Moreover, \(f'\) is always positive for \(x\lt-1\) and \(f'\) is always negative for \(x\gt-1\text{,}\) so a lowest point of \(f\) is never attained.

SubsectionGlobal Optimization

In both Figure3.18 and Example3.19 above, we were interested in finding the global minimum and global maximum for \(f\) on its entire domain. At other times, we might focus on some restriction of the domain.

For example, rather than considering \(f(x) = 2 + \frac{3}{1+(x+1)^2}\) for every value of \(x\text{,}\) perhaps instead we are only interested in those \(x\) for which \(0 \le x \le 4\text{,}\) and we would like to know which values of \(x\) in the interval \([0,4]\) produce the largest possible and smallest possible values of \(f\text{.}\) We are accustomed to critical numbers playing a key role in determining the location of extreme values of a function; now, by restricting the domain to an interval it makes sense that the endpoints of the interval will also be important to consider, as we see in the following example. When limiting ourselves to a particular interval, we will often refer to the absolute maximum or minimum value, rather than the global maximum or minimum.

Example3.20

Let \(g(x) = \frac{1}{3}x^3 - 2x + 2\text{.}\)

Find all critical numbers of \(g\) that lie in the interval \(-2 \le x \le 3\text{.}\)

Use a graphing utility to construct the graph of \(g\) on the interval \(-2 \le x \le 3\text{.}\)

From the graph, determine the \(x\)-values at which the absolute minimum and absolute maximum of \(g\) occur on the interval \([-2,3]\text{.}\)

How do your answers change if we instead consider the interval \(-2 \le x \le 2\text{?}\)

What if we instead consider the interval \(-2 \le x \le 1\text{?}\)

Below is the graph of \(y=g(x)\) replicated three times, with critical points and endpoints of each interval marked. On the left, we have the interval \([-2,3]\) for (c); the middle shows the interval \([-2,2]\) for (d); the right displays the interval \([-2,1]\) for (e).

On \([-2,3]\text{,}\) \(g\) has a global maximum at \(x = 3\) and a global minimum at \(x = \sqrt{2}\text{.}\)

On \([-2,2]\text{,}\) \(g\) has a global maximum at \(x = -\sqrt{2}\) and a global minimum at \(x = \sqrt{2}\text{.}\)

On \([-2,3]\text{,}\) \(g\) has a global maximum at \(x = -\sqrt{2}\) and a global minimum at \(x = 1\text{.}\)

Since \(g'(x) = x^2 - 2\text{,}\) the critical numbers of \(g\) are \(x = \pm \sqrt{2} \approx \pm 1.414\text{,}\) both of which lie in the interval \(-2 \le x \le 3\text{.}\)

Below is the graph of \(y=g(x)\) replicated three times, with critical points and endpoints of each interval marked. On the left, we have the interval \([-2,3]\) for (c); the middle shows the interval \([-2,2]\) for (d); the right displays the interval \([-2,1]\) for (e).

On \([-2,3]\text{,}\) \(g\) has a global maximum at \(x = 3\) and a global minimum at \(x = \sqrt{2}\text{.}\)

On \([-2,2]\text{,}\) \(g\) has a global maximum at \(x = -\sqrt{2}\) and a global minimum at \(x = \sqrt{2}\text{.}\)

On \([-2,3]\text{,}\) \(g\) has a global maximum at \(x = -\sqrt{2}\) and a global minimum at \(x = 1\text{.}\)

Example3.20 showed how the absolute maximum and absolute minimum of a function on a closed, bounded interval \([a,b]\) depend not only on the critical numbers of the function, but also on the values of \(a\) and \(b\text{.}\) In fact, the interval we choose has nearly the same influence on extreme values as the function under consideration. For instance, Figure3.21 below depicts again the graph of \(y=g(x)\) that we looked at in Example3.20.

From left to right, the interval under consideration is changed from \([-2,3]\) to \([-2,2]\) to \([-2,1]\text{.}\)

There are two critical numbers on the interval \([-2,3]\text{;}\) the absolute minimum is at one critical number and the absolute maximum is at the right endpoint.

On the interval \([-2,2]\text{,}\) both critical numbers are again in the interval. One critical value is still the absolute minimum; however, the absolute maximum is now at the other critical number rather than at the right endpoint.

Only one critical number lies on the interval \([-2,1]\text{.}\) That critical number gives the absolute maximum; the absolute minimum is at the right endpoint.

These observations demonstrate several important facts that hold more generally.

The Extreme Value Theorem

If \(f\) is a continuous function on a closed and bounded interval \([a,b]\text{,}\) then \(f\) attains both an absolute minimum and absolute maximum on \([a,b]\text{.}\) That is, there is some value \(x_m\) in \([a,b]\) such that \(f(x_m) \le f(x)\) for all \(x\) in \([a,b]\text{.}\) Similarly, there is a value \(x_M\) in \([a,b]\) such that \(f(x_M) \ge f(x)\) for all \(x\) in \([a,b]\text{.}\) Letting \(m = f(x_m)\) and \(M = f(x_M)\text{,}\) it follows that \(m \le f(x) \le M\) for all \(x\) in \([a,b]\text{.}\)

The Extreme Value Theorem tells us that on any closed and bounded interval \([a,b]\text{,}\) a continuous function has to achieve both an absolute minimum and an absolute maximum. The theorem does not tell us where these extreme values occur, only that they must exist. As we saw in Example3.20, the only possible locations for relative extrema are at the endpoints of the interval or at a critical number. It is important to consider only the critical numbers that lie within the interval.

We now have the following approach to finding the absolute maximum and minimum of a continuous function \(f\) on the interval \([a,b]\text{:}\)

Find all critical numbers of \(f\) that lie in the interval.

Evaluate the function \(f\) at each critical number in the interval and at each endpoint of the interval.

From among these function values, the smallest is the absolute minimum of \(f\) on the interval, while the largest is the absolute maximum.

Example3.22

Find the exact absolute maximum and minimum of each function on the stated interval.

\(h(x) = xe^{-x}\) on \([0,3]\)

\(p(t) = \sin(t) + \cos(t)\) on \([-\frac{\pi}{2}, \frac{\pi}{2}]\)

\(q(x) = \frac{x^2}{x-2}\) on \([3,7]\)

\(f(x) = 4 - e^{-(x-2)^2}\) on \((-\infty, \infty)\)

\(h(x) = xe^{-ax}\) on \(\left[0, \frac{2}{a}\right]\text{;}\) assume \(a \gt 0\)

\(f(x) = b - e^{-(x-a)^2}\) on \((-\infty, \infty)\text{;}\) assume \(a, b \gt 0\)

For \(h(x) = xe^{-x}\text{,}\) we know that \(h'(x) = e^{-x}+xe^{-x}(-1) = e^{-x}(1-x)\text{.}\) Therefore the only critical number of \(h\) is \(x = 1\text{.}\) Next, we compute \(h(1)\text{,}\) \(h(0)\text{,}\) and \(h(3)\text{.}\) Observe that

\(h(0) = 0\)

\(h(1) = e^{-1} \approx 0.36788\)

\(h(3) = 3e^{-3} \approx 0.14936\)

Thus, on \([0,3]\text{,}\) the absolute maximum of \(h\) is \(e^{-1}\) and the absolute minimum is \(0\text{.}\)

Given \(p(t) = \sin(t) + \cos(t)\text{,}\) it follows that \(p'(t) = \cos(t) - \sin(t)\text{,}\) so \(p'(t) = 0\) implies that \(\cos(t) =\sin(t)\text{.}\) The sine and cosine functions have the same value at \(\frac{\pi}{4} \pm k\pi\) for any integer \(k\text{.}\) The only time this occurs in \([-\frac{\pi}{2}, \frac{\pi}{2}]\) is at \(x = \frac{\pi}{4}\text{,}\) thus this is the only critical number of \(p\) in the given interval. Now

Hence the critical numbers of \(q\) are \(x = 0\) and \(x = 4\text{.}\) Only the latter critical number lies in the interval \([3,7]\text{,}\) and thus we evaluate \(q\) and find

\(q(3) = \frac{9}{1} = 9\)

\(q(4) = \frac{16}{2} = 8\)

\(q(7) = \frac{49}{5} = 9.8\)

We now see that on \([3,7]\) the absolute maximum of \(q\) is \(9.8\) and the absolute minimum is \(8\text{.}\)

We first observe that we are working on the domain of all real numbers rather than on a closed bounded interval. Hence the extreme value theorem does not apply and we need to think about the overall behavior of the function. First, since \(f(x) = 4 - e^{-(x-2)^2}\text{,}\) we see by the chain rule that \(f'(x) = -e^{-(x-2)^2}(-2(x-2)) = 2(x-2)e^{-(x-2)^2}\text{.}\) Since \(e^{-(x-2)^2}\) is always positive (and in particular is never zero), it follows that the only critical number of \(f\) is \(x = 2\text{.}\) Furthermore, with \(f'(x) = 2(x-2)e^{-(x-2)^2}\text{,}\) we see that \(f'(x) \lt 0\) for \(x \lt 2\text{,}\) and \(f'(x) \gt 0\) for \(x \gt 2\text{.}\) Thus \(f\) is decreasing for \(x \lt 2\) and increasing for \(x \gt 2\text{.}\) The first derivative test then tells us that \(f\) has an absolute minimum at \(x = 2\) and \(f\) does not have an absolute maximum.

This will be similar to part (a). We start by differentiating \(h\text{,}\) finding \(h'(x)=e^{-ax}(1-ax)\text{.}\) Thus the only critical number of \(h\) is \(\frac1a\text{,}\) which is on the interval \(\left[0,\frac2a\right]\text{.}\) Evaluating \(h\) at the interval endpoints and the critical number, we find

Thus on \(\left[0,\frac2a\right]\text{,}\) the absolute maximum of \(h\) is \(\frac1ae^{-1}=(ae)^{-1}\) and the absolute minimum is \(0\text{.}\)

This is similar to part (d), including the fact that we are looking at all real numbers and not a closed bounded domain. We first find that \(f'(x)=2(x-a)e^{-(x-a)^2}\) and the only critical number of \(f\) is \(x=a\text{.}\) Since \(f'(x)\lt0\) for \(x\lt a\) and \(f'(x)\gt0\) for \(x\gt a\text{,}\) the first derivative test tells us that \(f\) has a minimum at \(x=a\) but has no maximum. Finally, the value of the absolute minimum is \(f(a)=b-1\text{.}\)

SubsectionMoving Toward Applications

We conclude this section with an example of an applied optimization problem. It highlights the role that a closed, bounded domain can play in finding absolute extrema.

Example3.23

A 20 cm piece of wire is cut into two pieces. One piece is used to form a square and the other to form an equilateral triangle. How should the wire be cut to maximize the total area enclosed by the square and triangle? To minimize the area?

The area is maximized when the wire is uncut and all used for the square; the area is minimized when approximately \(11.3\) cm of wire are used for the triangle and approximately \(8.7\) cm are used for the square.

We begin by sketching a picture that illustrates the situation. The variable in the problem is where we decide to cut the wire, so we label the cut point at a distance \(x\) cm from one end of the wire and note that the remaining portion of the wire then has length \(20-x\) cm.

Figure3.24 below shows how the wire is used to form the two regular polygons. From the \(x\) cm segment of wire, we obtain an equilateral triangle with three sides of length \(\frac{x}{3}\text{;}\) the square shaped from the remaining \(20-x\) cm of wire will have four sides of length \(\frac{20-x}{4}\text{.}\)

At this point, we note that there are obvious restrictions on \(x\text{:}\) in particular, \(0 \le x \le 20\text{.}\) In the extreme cases, all of the wire is being used to make just one figure. For instance, if \(x = 0\) then all 20 cm of wire are used to make a square that is \(5 \times 5\text{.}\)

Now, our overall goal is to find the minimum and maximum areas that can be enclosed. Because the height \(h\) of an equilateral triangle is \(\sqrt{3}\) times half the length of the base \(b\text{,}\) the area of the triangle is

When we set \(A'(x) = 0\text{,}\) we find that \(x = \frac{180}{4\sqrt{3}+9} \approx 11.3007\) is the only critical number of \(A\) in the interval \([0,20]\text{.}\)

Evaluating \(A\) at the critical number and endpoints, we see that

Thus, the absolute minimum occurs when \(x \approx 11.3007\) cm and results in the minimum area of approximately \(10.8741\) square centimeters. The absolute maximum occurs when we invest all of the wire in the square (and none in the triangle), resulting in 25 square centimeters of area. These results are confirmed by a plot of \(y = A(x)\) on the interval \([0,20]\text{,}\) as shown below in Figure3.25.

Example3.26

A piece of cardboard that is \(10 \times 15\) (each measured in inches) is being made into a box without a top. To do so, squares are cut from each corner of the box and the remaining sides are folded up. Assuming that the box needs to be at least 1 inch deep and no more than 3 inches deep, answer the following questions.

Draw a labeled diagram that shows the given information. What variable should we introduce to represent the choice we make in creating the box? Label the diagram appropriately with the variable and write a sentence to state what the variable represents.

Determine a formula for the function \(V\) that tells us the volume of the box and depends on the variable defined in (a).

What is the domain of the function \(V\text{?}\) That is, in the context of this problem, what values of \(x\) make sense for the input of \(V\text{?}\) Are there additional restrictions provided in the problem?

Determine all critical numbers of the function \(V\text{.}\)

Evaluate \(V\) at each of the endpoints of the domain and at any critical numbers that lie in the domain.

What is the maximum possible volume of the box? What is the minimum volume? Justify your answers using calculus.

From each corner of the cardboard, we will cut a square with side length \(x\) inches. The result is the following picture:

After cutting out the four \(x\times x\) squares from the corners of the cardboard and folding up the edges, the resulting box is \(x\) inches tall, \(10-2x\) inches wide, and \(15-2x\) inches long. Thus the volume of the box (in cubic inches) is given by

At first glance, we see that \(V\) is a polynomial and so is defined for every real number \(x\text{.}\) However, because \(x\) represents a length in this problem, the smallest \(x\) can be is \(0\text{.}\) Moreover, since the shortest side of the cardboard is initially only \(10\) inches and cutting the squares reduces each side by \(2x\) inches, the largest \(x\) can be is \(5\text{.}\) We note that both of these extremes would result in a box with zero volume, as the first would have no height and the second would have no width. Finally, the problem says the height needs to be between \(1\) and \(3\) inches, so we will further restrict the domain of \(V\) to \(1 \le x \le 3\text{.}\)

Since \(V'(x) = 12x^2 - 100x + 150\text{,}\) it follows that the critical numbers (where \(V'(x) = 0\)) are

\begin{equation*}
x = \frac{25 \pm 5\sqrt{7}}{6} \approx 6.371, 1.962\text{.}
\end{equation*}

Only the latter critical number is in the relevant domain of \(V\text{,}\) so we consider

The maximum possible volume of the box is about \(132.038\) cubic inches, occurring when squares with side length around \(1.962\) inches are cut from each corner of the cardboard. The minimum possible volume of the box is \(104\) in\(^3\) and occurs with a box that is \(1\) inch tall.

Example3.23 and Example3.26 illustrate standard steps that we undertake in almost every applied optimization problem: we draw a picture to demonstrate the situation, introduce one or more variables to represent quantities that are changing, find a function that models the quantity to be optimized, and then decide on an appropriate domain for that function. Once that is done, we are in the familiar situation of finding the absolute minimum and maximum of a function over a particular domain, so we apply the calculus ideas that we have been studying up to this point in Chapter3.

SubsectionSummary

To find relative extreme values of a function, we use a first derivative sign chart and classify all of the function's critical numbers. If instead we are interested in absolute extreme values, we first decide whether we are considering the entire domain of the function or a particular interval.

In the case of finding global extrema over the function's entire domain, we again use a first or second derivative sign chart. If we are working to find absolute extrema on a restricted interval, then we first identify all critical numbers of the function that lie in the interval.

For a continuous function on a closed, bounded interval, the only possible points at which absolute extreme values occur are the critical numbers and the endpoints. Hence we evaluate the function at each endpoint and at each critical number in the interval and then compare the results to decide which is largest (the absolute maximum) and which is smallest (the absolute minimum).

Based on the given information about each function, decide whether the function has a global maximum, a global minimum, neither, both, or that it is not possible to say without more information. Assume that each function is twice differentiable and defined for all real numbers, unless noted otherwise. In each case, write one sentence to explain your conclusion.

\(f\) is a function such that \(f''(x) \lt 0\) for every \(x\text{.}\)

\(g\) is a function with two critical numbers \(a\) and \(b\) (where \(a \lt b\)), and \(g'(x) \lt 0\) for \(x \lt a\text{,}\) \(g'(x) \lt 0\) for \(a \lt x \lt b\text{,}\) and \(g'(x) \gt 0\) for \(x \gt b\text{.}\)

\(h\) is a function with two critical numbers \(a\) and \(b\) (where \(a \lt b\)), and \(h'(x) \lt 0\) for \(x \lt a\text{,}\) \(h'(x) \gt 0\) for \(a \lt x \lt b\text{,}\) and \(h'(x) \lt 0\) for \(x \gt b\text{.}\) In addition, \(\lim_{x \to \infty} h(x) = 0\) and \(\lim_{x \to -\infty} h(x) = 0\text{.}\)

\(p\) is a function differentiable everywhere except at \(x = a\) and \(p''(x) \gt 0\) for \(x \lt a\) and \(p''(x) \lt 0\) for \(x \gt a\text{.}\)

For each family of functions that depends on one or more parameters, determine the function's absolute maximum and absolute minimum on the given interval.

For each of the functions described below (each continuous on \([a,b]\)), state the location of the function's absolute maximum and absolute minimum on the interval \([a,b]\text{,}\) or say there is not enough information provided to make a conclusion. Assume that any critical numbers mentioned in the problem statement represent all of the critical numbers the function has in \([a,b]\text{.}\) In each case, write one sentence to explain your answer.

\(f'(x) \le 0\) for all \(x\) in \([a,b]\text{.}\)

\(g\) has a critical number at \(c\) such that \(a \lt c\lt b\) and \(g'(x) \gt 0\) for \(x \lt c\) and \(g'(x) \lt 0\) for \(x \gt c\text{.}\)

\(h(a) = h(b)\) and \(h''(x) \lt 0\) for all \(x\) in \([a,b]\text{.}\)

\(p(a) \gt 0\text{,}\) \(p(b) \lt 0\text{,}\) and for the critical number \(c\) such that \(a \lt c \lt b\text{,}\) \(p'(x) \lt 0\) for \(x \lt c\) and \(p'(x) \gt 0\) for \(x \gt c\text{.}\)

Let \(s(t) = 3\sin\left(2\left(t-\frac{\pi}{6}\right)\right) + 5\text{.}\) Find the exact absolute maximum and minimum of \(s\) on the provided intervals by testing the endpoints and by finding and evaluating \(s\) at all relevant critical numbers of \(s\text{.}\)