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Section 11.4 Extreme Values (AD4)

Subsection 11.4.1 Activities

Remark 11.39.

In many different settings, we are interested in knowing where a function achieves its least and greatest values. These can be important in applications—say to identify a point at which maximum profit or minimum cost occurs—or in theory to characterize the behavior of a function or a family of related functions.

Example 11.40.

Consider the familiar example of a parabolic function such as \(s(t) = -16t^2 + 32t + 48\text{.}\) This function represents the height of an object tossed vertically straight up: its maximum value occurs at the vertex of the parabola and represents the greatest height the object reaches. This maximum value is an especially important point on the graph and we can notice that the function changes from increasing to decreasing at this point.
Figure 118. The graph of \(s(t) = -16t^2 + 32t + 48\)

Definition 11.41.

We say that \(f(x)\) has a global maximum at \(x=c\) provided that \(f(c)\geq f(x)\) for all \(x\) in the domain of the function. We also say that \(f(c)\) is a global maximum value for the function. On the other hand, we say that \(f(x)\) has a global minimum at \(x=c\) provided that \(f(c)\leq f(x)\) for all \(x\) in the domain of the function. We also say that \(f(c)\) is a global minimum value for the function. The global maxima and minima are also known as the global extrema (or extreme values or absolute extrema) of the function.

Activity 11.42.

According to Definition 11.41, which of the following statements best describes the global extrema of the function in Figure 118?
  1. The global maximum is \(t = 1\text{,}\) because this is where the function goes from increasing to decreasing.
  2. The global maximum is \(s(1) = 64\text{,}\) because \(s(t)\leq 64\) for every other input \(t\text{.}\)
  3. The graph has two global minima at the endpoints because the endpoints must be global extrema.
  4. The graph has no global minimum.

Observation 11.43.

There can be some issues when trying to determine the global minimum and maximum values of a function only using its graph. The Extreme Value Theorem will guarantee the existence of global extrema on a closed interval. Then we will see how to use derivatives to find algebraically the extrema of a function.

Activity 11.45.

For each of the following figures, decide where the global extrema are located.
(a)
Figure 119.
(b)
Figure 120.
(c)
Figure 121.
(d)
Figure 122.

Activity 11.46.

The Extreme Value Theorem (EVT) guarantees a global maximum and a global minimum for which of the following?
  1. \(f(x)=\dfrac{x^{2}}{x^{2}-4x-5}\) on \([-5,0]\text{.}\)
  2. \(f(x)=\dfrac{x^{2}}{x^{2}-4x-5}\) on \([0,4]\text{.}\)
  3. \(f(x)=\dfrac{x^{2}}{x^{2}-4x-5}\) on \([4,6]\text{.}\)
  4. \(f(x)=\dfrac{x^{2}}{x^{2}-4x-5}\) on \([6,10]\text{.}\)

Activity 11.47.

For the following activity, draw a sketch of a function that has the following properties.
(a)
The function is continuous and has an global minimum but no global maximum.
(b)
The function is continuous and has an global maximum but no global minimum.

Definition 11.48.

We say that \(x=c\) is a critical point (or critical number) of \(f(x)\) if \(x=c\) is in the domain of \(f(x)\) and either \(f'(c) = 0\) or \(f'(c)\) does not exist.

Activity 11.49.

Which of the following are critical numbers for \(f(x) = \frac{1}{3}x^3 - 2x + 2\text{?}\)
  1. \(x = \sqrt{2}\) and \(x = -\sqrt{2}\text{.}\)
  2. \(x = \sqrt{2}\text{.}\)
  3. \(x = 2\) and \(x = 0\text{.}\)
  4. \(x = 2\text{.}\)

Remark 11.50. The Closed Interval Method.

The following is a way of finding the global extrema of a continuous function \(f\) on a closed interval \([a,b]\text{.}\)
  1. Make a list of all critical points of \(f\) in \((a,b)\text{.}\) (Do not include any critical points outside of the interval).
  2. Add the endpoints \(a\) and \(b\) to the list.
  3. Evaluate \(f\) at all points on your list.
  4. The smallest output occurs at the global minimum. The largest output occurs at the global maximum.

Activity 11.51.

What are the global extrema for \(f(x) = 3x^4 - 4x^3\) on \([-1,2]\text{.}\)
  1. Global maximum is when \(x = 0\) and global minimum when \(x = 1\text{.}\)
  2. Global maximum is when \(x = 2\) and global minimum when \(x = -1\text{.}\)
  3. Global maximum is when \(x = 2\) and global minimum when \(x = 1\text{.}\)
  4. Global maximum is when \(x = 0\) and global minimum when \(x = -1\text{.}\)

Activity 11.52.

What are the global extrema for \(f(x) = x\sqrt{4-x}\) on \([-2,4]\text{.}\)
  1. Global maximum is when \(x = -2\) and global minimum when \(x = \frac{8}{3}\text{.}\)
  2. Global maximum is when \(x = 4\) and global minimum when \(x = \frac{8}{3}\text{.}\)
  3. Global maximum is when \(x = \frac{8}{3}\) and global minimum when \(x = -2\text{.}\)
  4. Global maximum is when \(x = 4\) and global minimum when \(x = -2\text{.}\)

Activity 11.53.

Explain how to find the global minimum and global maximum values of the function \(f(x)=-2 \, x^{3} + 18 \, x^{2} + 42 \, x + 33\) on the interval \([-2,2]\text{.}\)

Activity 11.54.

In this problem you will consider the function \(g(x)\text{.}\)
\begin{equation*} g(x) = \left\{ \begin{array}{ll} x^3-3x & x \lt 0\\ x^2 -4x +2 & x\geq 0 \end{array} \right. \end{equation*}
(a)
What can you say about the point \(x=0\text{?}\)
(b)
In addition to \(x=0\text{,}\) find the other two critical points. What are the critical points of \(g(x)\text{?}\)
  1. \(\displaystyle x=0, \, x=1, \, x= 2\)
  2. \(\displaystyle x=0, \, x=-1, \, x= 2\)
  3. \(\displaystyle x=0, \, x=-1, \, x=-2\)
  4. \(\displaystyle x=0, \, x=1, \, x= -2\)
(c)
Can you use the Closed Interval Method on \([-4,-1]\text{?}\) If you can, find the global max and min. If you can’t, explain why.
(d)
Can you use the Closed Interval Method on \([1,4]\text{?}\) If you can, find the global max and min. If you can’t, explain why.
(e)
Can you use the Closed Interval Method on \([-1,1]\text{?}\) If you can, find the global max and min. If you can’t, explain why.

Subsection 11.4.2 Videos

Figure 123. Video for AD4

Subsection 11.4.3 Exercises

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