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  • Define absolute extrema.
  • Define local extrema.
  • Explain how to find the critical points of a function over a closed interval.
  • Describe how to use critical points to locate absolute extrema over a closed interval.

Given a particular function, we are often interested in determining the largest and smallest values of the function. This information is important in creating accurate graphs. Finding the maximum and minimum values of a function also has practical significance because we can use this method to solve optimization problems, such as maximizing profit, minimizing the amount of material used in manufacturing an aluminum can, or finding the maximum height a rocket can reach. In this section, we look at how to use derivatives to find the largest and smallest values for a function.

Absolute extrema

Consider the function f ( x ) = x 2 + 1 over the interval ( , ) . As x ± , f ( x ) . Therefore, the function does not have a largest value. However, since x 2 + 1 1 for all real numbers x and x 2 + 1 = 1 when x = 0 , the function has a smallest value, 1, when x = 0 . We say that 1 is the absolute minimum of f ( x ) = x 2 + 1 and it occurs at x = 0 . We say that f ( x ) = x 2 + 1 does not have an absolute maximum (see the following figure).

The function f(x) = x2 + 1 is graphed, and its minimum of 1 is seen to be at x = 0.
The given function has an absolute minimum of 1 at x = 0 . The function does not have an absolute maximum.

Definition

Let f be a function defined over an interval I and let c I . We say f has an absolute maximum    on I at c if f ( c ) f ( x ) for all x I . We say f has an absolute minimum    on I at c if f ( c ) f ( x ) for all x I . If f has an absolute maximum on I at c or an absolute minimum on I at c , we say f has an absolute extremum    on I at c .

Before proceeding, let’s note two important issues regarding this definition. First, the term absolute here does not refer to absolute value. An absolute extremum may be positive, negative, or zero. Second, if a function f has an absolute extremum over an interval I at c , the absolute extremum is f ( c ) . The real number c is a point in the domain at which the absolute extremum occurs. For example, consider the function f ( x ) = 1 / ( x 2 + 1 ) over the interval ( , ) . Since

f ( 0 ) = 1 1 x 2 + 1 = f ( x )

for all real numbers x , we say f has an absolute maximum over ( , ) at x = 0 . The absolute maximum is f ( 0 ) = 1 . It occurs at x = 0 , as shown in [link] (b).

A function may have both an absolute maximum and an absolute minimum, just one extremum, or neither. [link] shows several functions and some of the different possibilities regarding absolute extrema. However, the following theorem, called the Extreme Value Theorem , guarantees that a continuous function f over a closed, bounded interval [ a , b ] has both an absolute maximum and an absolute minimum.

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 −1 at ±π, ±3π, … and absolute maximums of 1 at 0, ±2π, ±4π, …. In figure d, the piecewise function f(x) = 2 – x2 for 0 ≤ x < 2 and x – 3 for 2 ≤ x ≤ 4 is shown, with absolute maximum of 2 at x = 0 and no absolute minimum. In figure e, the function f(x) = (x – 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 − x) is shown on [0, 2), with absolute minimum of 0 at x = 0 and no absolute maximum.
Graphs (a), (b), and (c) show several possibilities for absolute extrema for functions with a domain of ( , ) . Graphs (d), (e), and (f) show several possibilities for absolute extrema for functions with a domain that is a bounded interval.

Extreme value theorem

If f is a continuous function over the closed, bounded interval [ a , b ] , then there is a point in [ a , b ] at which f has an absolute maximum over [ a , b ] and there is a point in [ a , b ] at which f has an absolute minimum over [ a , b ] .

Practice Key Terms 9

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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