Extrema and the First-Derivative Test

Transcription

1 Extrema and the First-Derivative Test MATH 151 Calculus for Management J. Robert Buchanan Department of Mathematics 2018

2 Why Maximize or Minimize? In almost all quantitative fields there are objective outcomes that we would like to optimize (either by maximizing or minimizing some quantity).

3 Why Maximize or Minimize? In almost all quantitative fields there are objective outcomes that we would like to optimize (either by maximizing or minimizing some quantity). A business owner wants to maximize revenue while minimizing cost.

4 Why Maximize or Minimize? In almost all quantitative fields there are objective outcomes that we would like to optimize (either by maximizing or minimizing some quantity). A business owner wants to maximize revenue while minimizing cost. An investor wishes to maximize return on investment while minimizing risk of loss of investment.

5 Why Maximize or Minimize? In almost all quantitative fields there are objective outcomes that we would like to optimize (either by maximizing or minimizing some quantity). A business owner wants to maximize revenue while minimizing cost. An investor wishes to maximize return on investment while minimizing risk of loss of investment. An electrical engineer must minimize power consumption by an electrical device.

6 Relative Extrema The increasing and decreasing information provided by the first derivative can help us find maximum and minimum values of a function.

7 Relative Extrema The increasing and decreasing information provided by the first derivative can help us find maximum and minimum values of a function. Definition 1. f (c) is a relative maximum of f if f (c) f (x) for all x in some open interval containing c. 2. f (c) is a relative minimum of f if f (c) f (x) for all x in some open interval containing c. In either case, we call f (c) a relative extremum.

8 Relative Extrema The increasing and decreasing information provided by the first derivative can help us find maximum and minimum values of a function. Definition 1. f (c) is a relative maximum of f if f (c) f (x) for all x in some open interval containing c. 2. f (c) is a relative minimum of f if f (c) f (x) for all x in some open interval containing c. In either case, we call f (c) a relative extremum. The terms local maximum, local minimum, and local extremum are used in some textbooks.

9 Where are Local Extrema Found? Consider the graph below and determine the types of points where an extremum may occur x 0.5

10 Locations of Local Extrema x 0.5

11 Locations of Local Extrema x 0.5 It seems that an extremum can occur where f (x) = 0 or f (x) is undefined.

12 Occurrences of Relative Extrema Theorem Suppose that f has a relative maximum or minimum at x = c, then c is a critical number of f. That is, either f (c) = 0 or f (c) is undefined.

13 First Derivative Test Theorem (First Derivative Test) Suppose that f is continuous on interval (a, b) and c is the only critical number. If f is differentiable on the interval (except possibly at c), then f (c) can be classified as a relative minimum, a relative maximum, or neither, as shown below. 1. If f (x) < 0 for x to the left of c and f (x) > 0 for x to the right of c [in other words f changes from decreasing to increasing at c], then f (c) is a relative minimum. 2. If f (x) > 0 for x to the left of c and f (x) < 0 for x to the right of c [in other words f changes from increasing to decreasing at c], then f (c) is a relative maximum. 3. If f (x) has the same sign on both sides of c, then f (c) is not a relative extremum.

14 Example (1 of 2) Example Find all the relative extrema for the function below. f (x) = x 4 + 6x 2 2

15 Example (1 of 2) Example Find all the relative extrema for the function below. f (x) = x 4 + 6x 2 2 f (x) = 4x x = 4x(x 2 + 3)

16 Example (2 of 2) x

17 Absolute Extrema There are situations in which we are interested in the greatest and the least values of a function. These must be global properties of the function, not local properties.

18 Absolute Extrema There are situations in which we are interested in the greatest and the least values of a function. These must be global properties of the function, not local properties. Definition Let f be a function defined on an interval I containing c. 1. f (c) is the absolute maximum of f on I if f (c) f (x) for all x in I. 2. f (c) is the absolute minimum of f on I if f (c) f (x) for all x in I. An absolute maximum or an absolute minimum is referred to as an absolute extremum.

19 Example (1 of 4) Example x 2 Suppose f (x) = and I = (1, ). Does f have an (x 1) 2 absolute maximum and absolute minimum on I? x

20 Example (2 of 4) Example x 2 Suppose f (x) = and I = [ 1, 1). Does f have an (x 1) 2 absolute maximum and absolute minimum on I? x

21 Example (3 of 4) Example x 2 Suppose f (x) = and I = [ 1, 1/2]. Does f have an (x 1) 2 absolute maximum and absolute minimum on I? x

22 Example (4 of 4) Example x 2 Suppose f (x) = and I = [ 1, 1/2). Does f have an (x 1) 2 absolute maximum and absolute minimum on I? x

23 Extreme Value Theorem Question: under what conditions will a function have an absolute maximum and absolute minimum?

24 Extreme Value Theorem Question: under what conditions will a function have an absolute maximum and absolute minimum? Theorem (Extreme Value Theorem) A continuous function f defined on a closed, bounded interval [a, b] attains both an absolute maximum and an absolute minimum on that interval.

25 Guidelines To find the extrema of a continuous function f on a closed, bounded interval [a, b], follow these steps: 1. Evaluate f at each of its critical numbers in (a, b). 2. Evaluate f at each endpoint a and b. 3. The least of these values is the absolute minimum, and the greatest of these values is the absolute maximum.

26 Example (1 of 2) Example Find the absolute extrema of f (x) = x 4 8x on the interval [ 1, 3].

27 Example (2 of 2) f (x) = x 4 8x f (x) = 4x(x 2 4) = 4x(x 2)(x + 2)

28 Example (2 of 2) f (x) = x 4 8x f (x) = 4x(x 2 4) = 4x(x 2)(x + 2) The critical points are c = 0 and c = 2 (note x = 2 is not in the interval [ 1, 3]).

29 Example (2 of 2) f (x) = x 4 8x f (x) = 4x(x 2 4) = 4x(x 2)(x + 2) The critical points are c = 0 and c = 2 (note x = 2 is not in the interval [ 1, 3]). f ( 1) = 5 f (0) = 2 f (2) = 14 (absolute minimum) f (3) = 11 (absolute maximum)

30 Example Example Find the absolute extrema of f (x) = x 3 3x 2 + 3x + 1 on the interval [0, 2] f x

31 Cost, Revenue, and Profit Analysis Example A marketing analyst for a company that produces skateboards has determined that if the company sells x units of the deluxe model, the revenue function is R(x) = 79.9x 0.03x 2 dollars and the total cost function is C(x) = 0.08x x dollars. If production is limited to 0 x 380, find the production level which will yield the maximum profit.

32 Homework Read Section 3.2 Exercises: 1, 5, 15, 23, 39, 47