Relative Extrema Michael Freeze MAT 151 UNC Wilmington Summer - - PowerPoint PPT Presentation

Relative Extrema

Michael Freeze

MAT 151 UNC Wilmington

Summer 2013

1 / 13

Section 5.2 :: Relative Extrema

2 / 13

Relative Maximum or Minimum

Let c be a number in the domain of a function f . Then f (c) is a relative (or local) maximum for f if there exists an open interval (a, b) containing c such that f (x) ≤ f (c) for all x in (a, b). Likewise, f (c) is a relative (or local) minimum for f if there exists an open interval (a, b) containing c such that f (x) ≥ f (c) for all x in (a, b).

3 / 13

Relative Maximum or Minimum

−2 2 4 6 −20 20

4 / 13

Fermat’s Theorem

If a function f has a relative extremum at c, then c is a critical number or c is an endpoint of the domain. −4 −2 2 4 6 −20 −10 10 20 30

5 / 13

First Derivative Test

Let c be a critical number for a function f . Suppose that f is continuous on (a, b) and differentiable on (a, b) except possibly at c, and that c is the only critical number for f in (a, b).

1 f (c) is a relative maximum of f if the derivative f ′(x) is

positive in the interval (a, c) and negative in the interval (c, b).

2 f (c) is a relative minimum of f if the derivative f ′(x) is

negative in the interval (a, c) and positive in the interval(c, b).

Example

Find the location of each relative extremum of the function. f (x) = x2 + 8x + 5

6 / 13

First Derivative Test

Let c be a critical number for a function f . Suppose that f is continuous on (a, b) and differentiable on (a, b) except possibly at c, and that c is the only critical number for f in (a, b).

1 f (c) is a relative maximum of f if the derivative f ′(x) is

positive in the interval (a, c) and negative in the interval (c, b).

2 f (c) is a relative minimum of f if the derivative f ′(x) is

negative in the interval (a, c) and positive in the interval(c, b).

Example

Find the location of each relative extremum of the function. f (x) = x3 + 3x2 − 24x + 2

7 / 13

First Derivative Test

Let c be a critical number for a function f . Suppose that f is continuous on (a, b) and differentiable on (a, b) except possibly at c, and that c is the only critical number for f in (a, b).

1 f (c) is a relative maximum of f if the derivative f ′(x) is

positive in the interval (a, c) and negative in the interval (c, b).

2 f (c) is a relative minimum of f if the derivative f ′(x) is

negative in the interval (a, c) and positive in the interval(c, b).

Example

Find the location of each relative extremum of the function. f (x) = x4 − 18x2 − 4

8 / 13

First Derivative Test

Let c be a critical number for a function f . Suppose that f is continuous on (a, b) and differentiable on (a, b) except possibly at c, and that c is the only critical number for f in (a, b).

1 f (c) is a relative maximum of f if the derivative f ′(x) is

positive in the interval (a, c) and negative in the interval (c, b).

2 f (c) is a relative minimum of f if the derivative f ′(x) is

negative in the interval (a, c) and positive in the interval(c, b).

Example

Find the location of each relative extremum of the function. f (x) = x2 − 2x + 1 x − 3

9 / 13

First Derivative Test

Let c be a critical number for a function f . Suppose that f is continuous on (a, b) and differentiable on (a, b) except possibly at c, and that c is the only critical number for f in (a, b).

1 f (c) is a relative maximum of f if the derivative f ′(x) is

positive in the interval (a, c) and negative in the interval (c, b).

2 f (c) is a relative minimum of f if the derivative f ′(x) is

negative in the interval (a, c) and positive in the interval(c, b).

Example

Find the location of each relative extremum of the function. f (x) = x2 − 6x + 9 x + 2

10 / 13

Attitude Change

Social psychologists have found that as the discrepancy between the views of a speaker and those of an audience increases, the attitude change in the audience also increases to a point but decreases when the discrepancy becomes too large, particularly if the communicator is viewed by the audience as having low credibility. Suppose that the degree of change can be approximated by the function D(x) = −x4 + 8x3 + 80x2, where x is the discrepancy between the views of the speaker and those of the audience, as measured by scores on a

• questionnaire. Find the amount of discrepancy the speaker

should aim for to maximize the attitude change in the audience. Source: Journal of Personality and Social Psychology.

11 / 13

Film Length

A group of researchers found that people prefer training films of moderate length; shorter films contain too little information, while longer films are

• boring. For a training film on the care of exotic

birds, the researchers determined that the ratings people gave for the film could be approximated by R(t) = 20t t2 + 100, where t is the length of the film (in minutes). Find the film length that received the highest rating.

12 / 13

Height

After a great deal of experimentation, two Atlantic Institute of Technology senior physics majors determined that when a bottle of French champagne is shaken several times, held upright, and uncorked, its cork travels according to s(t) = −16t2 + 40t + 3, where s is its height (in feet) above the ground t seconds after being released. (a) How high will it go? (b) How long is it in the air?

13 / 13