Section 4.3 Maxima and Minima
Learning Objectives.
Define absolute extrema.
Define local extrema.
Explain how to find the critical points of a function over an interval.
Describe how to use critical points to locate absolute extrema over an interval.
Subsection 4.3.1 Absolute Extrema
Consider the function
Definition 4.25.
Let
!["This figure has six parts a, b, c, d, e, and f. In figure a, the line f(x) = x3 is shown, and it is noted that it has no absolute minimum and no absolute maximum. In figure b, the line f(x) = 1/(x2 + 1) is shown, which is near 0 for most of its length and rises to a bump at (0, 1); it has no absolute minimum, but does have an absolute maximum of 1 at x = 0. In figure c, the line f(x) = cos x is shown, which has absolute minimums of β1 at Β±Ο, Β±3Ο, β¦ and absolute maximums of 1 at 0, Β±2Ο, Β±4Ο, β¦. In figure d, the piecewise function f(x) = 2 β x2 for 0 β€ x \lt 2 and x β 3 for 2 β€ x β€ 4 is shown, with absolute maximum of 2 at x = 0 and no absolute minimum. In figure e, the function f(x) = (x β 2)2 is shown on [1, 4], which has absolute maximum of 4 at x = 4 and absolute minimum of 0 at x = 2. In figure f, the function f(x) = x/(2 β x) is shown on [0, 2), with absolute minimum of 0 at x = 0 and no absolute maximum."](external/CNX_Calc_Figure_04_03_010.jpg)
Theorem 4.27. Extreme Value Theorem.
If
Subsection 4.3.2 Local Extrema and Critical Points
Consider the function
Definition 4.29.
A function
Definition 4.30.
Let
Theorem 4.31. Fermatβs Theorem.
If

Example 4.33. Locating Critical Points.
For each of the following functions, find all critical points. Use a graphing utility to determine whether the function has a local extremum at each of the critical points.
- The derivative
is defined for all real numbers Therefore, we only need to find the values for where Since the critical points are and From the graph of in Figure 4.34, we see that has a local maximum at and a local minimum atFigure 4.34. This function has a local maximum and a local minimum. - Using the chain rule, we see the derivative is
has critical points when and when We conclude that the critical points are From the graph of in Figure 4.35, we see that has a local (and absolute) minimum at but does not have a local extremum at orFigure 4.35. This function has three critical points: and The function has a local (and absolute) minimum at but does not have extrema at the other two critical points. - By the chain rule, we see that the derivative is
where Solving we see that which implies Therefore, the critical points are From the graph of in Figure 4.36, we see that has an absolute maximum at and an absolute minimum at Hence, has a local maximum at and a local minimum at (Note that if has an absolute extremum over an interval at a point that is not an endpoint of then has a local extremum atFigure 4.36. This function has an absolute maximum and an absolute minimum.
Checkpoint 4.37.
Find all critical points for
Subsection 4.3.3 Locating Absolute Extrema
The extreme value theorem states that a continuous function over a closed, bounded interval has an absolute maximum and an absolute minimum. As shown in Figure 4.26, one or both of these absolute extrema could occur at an endpoint. If an absolute extremum does not occur at an endpoint, however, it must occur at an interior point, in which case the absolute extremum is a local extremum. Therefore, by Theorem 4.31, the pointTheorem 4.38. Location of Absolute Extrema.
Let
Note 4.39. Problem-Solving Strategy: Locating Absolute Extrema over a Closed Interval.
Consider a continuous function
Evaluate
at the endpoints andFind all critical points of
that lie over the interval and evaluate at those critical points.Compare all values found in (1) and (2). From Theorem 4.38, the absolute extrema must occur at endpoints or critical points. Therefore, the largest of these values is the absolute maximum of
The smallest of these values is the absolute minimum of
Example 4.40. Locating Absolute Extrema.
For each of the following functions, find the absolute maximum and absolute minimum over the specified interval and state where those values occur.
over over
- Step 1. Evaluate
at the endpoints and is defined for all real numbers Therefore, there are no critical points where the derivative is undefined. It remains to check where Since at and is in the interval is a candidate for an absolute extremum of over We evaluate and findFrom the table, we find that the absolute maximum ofTable 4.41. Conclusion Absolute maximum Absolute minimum over the interval [1, 3] is and it occurs at The absolute minimum of over the interval [1, 3] is and it occurs at as shown in the following graph.Figure 4.42. This function has both an absolute maximum and an absolute minimum. - Step 1. Evaluate
at the endpoints and is given by The derivative is zero when which implies The derivative is undefined at Therefore, the critical points of are The point is an endpoint, so we already evaluated in step 1. The point is not in the interval of interest, so we need only evaluate We find thatWe conclude that the absolute maximum ofTable 4.43. Conclusion Absolute maximum Absolute minimum over the interval [0, 2] is zero, and it occurs at The absolute minimum is -2, and it occurs at as shown in the following graph.Figure 4.44. This function has an absolute maximum at an endpoint of the interval.
Checkpoint 4.45.
Find the absolute maximum and absolute minimum of

Note 4.47. Problem-Solving Strategy: Locating and Determining the Existence of Absolute Extrema over an Interval that is not Closed.
Consider a continuous function
-
We would like to evaluate
at the endpoints and but they may not be included in the interval. As a result, we may need to take a one sided limit. The proceedure is outlined below:If the left endpoint
is not included in the interval (i.e. the given interval is (a,b]), then evaluate andIf the right endpoint
is not included in the interval (i.e. the given interval is [a,b)), then evaluate andIf both the left and right endpoints
and are not included in the interval (i.e. the given interval is (a,b)), then evaluate and Find all critical points of
that lie over the interval and evaluate at those critical points.Compare all values found in (1) and (2). We know that the absolute extrema must occur at endpoints or critical points. However, if absolute extrema occurs at an endpoint not included in the interval, we say the absolute maximum or absolute minimum DNE. Therefore, the largest of these values is the absolute maximum of
if it occurred at a point included in the interval. The smallest of these values is the absolute minimum of if it occurred at a point included in the interval. Otherwise, the absolute maximum or absolute minimum DNE.
Example 4.48.
Find the absolute maximum and absolute minimum of
[-7,1]
(-7,1]
[-7,1)
- For the interval From the table, we find that the absolute maximum of
Table 4.49. Conclusion Absolute minimum Absolute maximum over the interval [-7,1] is and it occurs at The absolute minimum of over the interval [-7, 1] is and it occurs at as shown in the following graph.Figure 4.50. on the interval [-7,1]. - For the interval From the table, we find that the absolute maximum of
Table 4.51. Conclusion Absolute minimum Absolute maximum over the interval (-7,1] is and it occurs at The absolute minimum of over the interval (-7, 1] is and it occurs at as shown in the following graph.Figure 4.52. on the interval (-7,1]. - For the interval From the table, we find that the absolute maximum of
Table 4.53. Conclusion Absolute minimum Absolute maximum DNE over the interval [-7,1) DNE since is not included in the interval. The absolute minimum of over the interval [-7, 1) is and it occurs at as shown in the following graph.Figure 4.54. on the interval [-7,1).
Checkpoint 4.55.
Find the absolute maximum and absolute minimum of
Subsection 4.3.4 Key Concepts
A function may have both an absolute maximum and an absolute minimum, have just one absolute extremum, or have no absolute maximum or absolute minimum.
If a function has a local extremum, the point at which it occurs must be a critical point. However, a function need not have a local extremum at a critical point.
A continuous function over a closed, bounded interval has an absolute maximum and an absolute minimum. Each extremum occurs at a critical point or an endpoint. If the interval is not closed, then there may not exist an absolute maximum or minimum.