5.2 relative extrema find relative extrema of a function using the first derivative copyright ©...

5.2 Relative Extrema

• Find Relative Extrema of a function using the first derivative

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-

Wesley

If a function f has a relative extreme value f (c) on an open interval; then c is a critical value. So,

f (c) = 0 or f (c) does not exist.

Relative extrema will only occur at points where the derivative is = 0 or where it is undefined.

Relative Extrema

Relative Minimum

Let I be the domain of f :f (c) is a relative minimum (bottom of a valley) if

there exists within I an open interval I1 containing c such that f (c) ≤ f (x) for all x in I1;

f has a relative minimum at c if f (x) < 0 on(a, c) and f (x) > 0 on (c, b). That is, f is decreasing to the left of c and increasing to the right of c.

If the graph is continuous (no break) at the point where the function changes from decreasing to increasing, that point is called a relative minimum point

Relative Maximum

Let I be the domain of f :F (c) is a relative maximum (top of a hill) if

there exists within I an open interval I2 containing c such that f (c) ≥ f (x) for all x in I2.

f has a relative maximum at c if f (x) > 0 on(a, c) and f (x) < 0 on (c, b). That is, f is increasing to the left of c and decreasing to the right of c.

If the graph is continuous (no break) at the point where the function changes from increasing to decreasing, that point is called a relative maximum point.


WesleySlide 2.1- 7

Using First Derivatives to Find Maximum and Minimum Values

Graph over the interval (a, b)

f (c) Sign of f (x) for x in (a, c)

Sign of f (x) for x in (c, b)

Increasing or decreasing

Relative minimum

Relative maximum

–

+

+

–

Decreasing on (a, c];

increasing on [c, b)

Increasing on (a, c];

decreasing on [c, b)

c ba

c ba


WesleySlide 2.1- 8


Graph over the interval (a, b)

f (c) Sign of f (x) for x in (a, c)

Sign of f (x) for x in (c, b)

Increasing or decreasing

No relative maxima

or minima

No relative maxima

or minima

–

+

–

+

Decreasing on (a, b)

Increasing on (a, b)

c ba

c ba


WesleySlide 2.1- 9


Example 1: For the function f given by

find the relative extrema.

f (x) 2x3 3x2 12x 12.


WesleySlide 2.1- 10


Example 1 (continued): Find Derivative And set it = 0

These two critical values partition the number line into 3 intervals: A (– ∞, –1), B (–1, 2), and C (2, ∞).

CB A

2-1

6x2 6x 12 0

x2 x 2 0

(x 2)(x 1) 0

x 2 or x 1


WesleySlide 2.1- 11

Example 1 (continued):3rd analyze the sign of f (x) in each interval.

Test Value x = –2 x = 0 x = 4

Sign off (x)

+ – +

Resultf is increasing on (–∞, –1]

f is decreasing on [–1, 2]

f is increasing on [2, ∞)


xInterval

CB A

2-1


WesleySlide 2.1- 12


Example 1 (concluded):Therefore, by the First-Derivative Test,

f has a relative maximum at x = –1 given by

The relative maximum value is 19. It occurs where x = -1.

And f has a relative minimum at x = 2 given by

The relative minimum value is -8. It occurs where x is 2.

f ( 1) 2( 1)3 3( 1)2 12( 1)12 19

f (2) 2(2)3 3(2)2 12(2)12 8


WesleySlide 2.1- 13


Example 1 (continued):


WesleySlide 2.1- 14


Example 3: Find the relative extrema for the Function f (x) given by

Then sketch the graph.

1st find f (x).

f (x) (x 2)2 3 1

f (x) 2

3x 2

1 3

f (x) 2

3 x 23


WesleySlide 2.1- 15


Example 3 (continued): 2nd find where f (x) does not exist or where f (x) = 0.

Note that f (x) does not exist where the denominator equals 0. Since the denominator equals 0 when x = 2, x = 2 is a critical value.

f (x) = 0 where the numerator equals 0. Since 2 ≠ 0, f (x) = 0 has no solution.

Thus, x = 2 is the only critical value.


WesleySlide 2.1- 16


Example 3 (continued): 3rd x = 2 partitions the number line into 2 intervals:

A (– ∞, 2) and B (2, ∞). So, analyze the signs of f (x) in both intervals.

Test Value x = 0 x = 3

Sign of f (x) – +

Result f is decreasing

on (– ∞, 2]f is increasing on

[2, ∞)

xInterval

B A

2


WesleySlide 2.1- 17


Example 3 (continued):Therefore, by the First-Derivative Test,

f has a relative minimum at x = 2 given by

The relative minimum value is 1. It occurs at x = 2.

f (2) (2 2)2 3 1 1


WesleySlide 2.1- 18


Example 3 (concluded):We use the information obtained to sketch the graph below, plotting other function values as needed.


WesleySlide 2.1- 19

More Examples

3

2

.

( ) 1

( )4

4 21( )

2

Find the relativeextrema

f x x

xf x

x

x xf x

x


WesleySlide 2.1- 20

C(q) above is the cost function. p(q) is the price function.

Finda)The number of units that will produce a maximum profit.b)The maximum profit.c)The price that will produce a maximum profit.

C( ) 25q 5000

p(q) 90 0.02q

q

a) 1625 b) $47812.50 c) $57.50

5.2 relative extrema find relative extrema of a function using the first derivative copyright ©...

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