2.4 Absolute Maximum And Minimum Values

Math 135 Class Notes

2.4

Business Calculus

Spring 2009

Using Derivatives to Find Absolute Maximum and Minimum Values

ABSOLUTE MAXIMUM AND MINIMUM VALUES In many applications, we’re interested in finding the largest value, or absolute maximum, or the smallest value, or absolute minimum, of some quantity. DEFINITION OF ABSOLUTE EXTREMA Suppose that f is a function with domain I. • f (c) is an absolute minimum if f (c) ≤ f (x) for all x in I. So f (c) is the largest function value over the domain of f . • f (c) is an absolute maximum if f (c) ≥ f (x) for all x in I. So f (c) is the smallest function value over the domain of f . As illustrated in the following graph, the absolute maximum and absolute minimum of a function may occur at a relative extrema or at another point.

FINDING ABSOLUTE MAXIMUM AND MINIMUM VALUES OVER CLOSED INTERVALS The function in the above graph has both an absolute maximum, at the right endpoint b, and an absolute minimum, at a crticial value which is also a relative minimum. However, not every function has an absolute maximum and absolute minimum over its domain. 1 EXAMPLE Let f (x) = on the domain (0, 2). Does this function have an absolute maximum 2−x and/or an absolute minimum on its domain? y 4

3

2

1

0 0

1

2

3

x

57

58

Chapter 2

Applications of Differentiation

One of the “problems” with the function in the preceding example is that its domain is an open interval which does not include its endpoints. The following theorem tells us that for a continuous function whose domain is a closed interval, we’re guaranteed of having an absolute maximum and an absolute minimum. THEOREM 7 The Extreme-Value Theorem A continuous function f defined over a closed interval [a, b] must have an absolute maximum value and an absolute minimum value over [a, b]. We need both hypotheses in the theorem—continuity and a closed interval—for the conclusion of the theorem to hold. If either of these does not hold, then the function may not have absolute extrema. 1 For instance, the function f (x) = with domain (0, 2) in the above example does not have 2−x absolute extrema. In this case, the function is continous but the domain (0, 2) is not a closed inteval. The Extreme Value Theorem says that a continuous function on a closed interval has an absolute maximum value and an absolute minimum value. However, it does not tell us how to find these extreme values. In order to find the absolute extrema, we can use the following result. THEOREM 8 Maximum-Minimum Principle 1 Suppose that f is a continuous function defined over a closed interval [a, b]. Then the absolute maximum must occur either at a relative maximum or at an endpoint of the interval. The absolute minimum must occur either at a relative minimum or at an endpoint. To find the absolute maximum and minimum values over [a, b]: a) First find f 0 (x). b) Then determine all critical values in [a, b]. That is, find all c in [a, b] for which f 0 (c) = 0

or

f 0 (c) does not exist.

c) List the values from step (b) and the endpoints of the interval: a, c1 , c2 , . . . , cn , b. d) Evaluate f (x) for each value in step (c): f (a), f (c1 ), f (c2 ), . . . , f (cn ), f (b). The largest of these is the absolute maximum of f over [a, b]. The smallest of these is the absolute minimum of f over [a, b]. EXAMPLE [−2, 32 ].

Find the absolute maximum and minimum values of f (x) = x3 − 3x + 2 over the interval

2.4

59

Using Derivatives to Find Absolute Maximum and Minimum Values

FINDING ABSOLUTE MAXIMUM AND MINIMUM VALUES OVER OTHER INTERVALS When a function has only one critical value, then the absolute maximum or absolute minimum will occur at that critical value provided that it is a relative maximum or minimum. We can use the Second-Derivative Test to determine whether the function has a relative maximum or minimum at the crtical value. This works whether the domain is a closed interval or an open interval. THEOREM 9 Maximum-Minimum Principle 2 Suppose that f is a function such that f 0 (x) exists for every x in an interval I, and that there is exactly one (critical) value c in I, for which f 0 (c) = 0. Then • f (c) is the absolute maximum value over I if f 00 (c) < 0. • f (c) is the absolute minimum value over I if f 00 (c) > 0. EXAMPLE

Find the absolute maximum and absolute minimum values, if they exist, of the function f (x) = 5x − 2x2 .

Note that when no domain is specified, then it’s assumed to be (−∞, ∞), the set of all real numbers. y 4 3 2 1

–2

–1

1

2

3

4

x

–1 –2

Combining the results of the preceding discussion, we can state the following general strategy for finding maximum and minimum values. A STRATEGY FOR FINDING ABSOLUTE MAXIMUM AND MINIMUM VALUES To find absolute maximum and minimum values of a continuous function over an interval: a) Find f 0 (x). b) Find the critical values. c) If the interval is closed, use Maximum-Minimum Principle 1. d) If the interval is not closed, such as (−∞, ∞), (0, ∞), or (a, b), and the function has only one critical value, use Maximum-Minimum Principle 2. In such a case, if the function has a maximum, it will have no minimum; and if it has a minimum, it will have no maximum. To show that a function does not have an absolute maximum or an absolute minimum, look at the values of the function as x approaches the endpoints of the interval, or as x approaches ±∞.

60

Chapter 2

Applications of Differentiation

35 . x

EXAMPLE

Find the absolute maximum and minimum values, if they exist, of f (x) = 5x +

EXAMPLE

Find the absolute maximum and minimum values, if they exist, of f (x) = (x − 2)3 + 1.