increasing and decreasing functions and the first derivative test

Increasing and Decreasing Functions and the First Derivative Test

Determine the intervals on which a function is increasing or decreasing Apply the First Derivative Test to find relative extrema of a function

Standard 4.5a

Incr

easin

g

Constant

Decreasing

y

x

Test for Increasing and Decreasing Functions

Let f be differentiable on the interval (a, b)

1. If f’(x) > 0 then f is increasing on (a, b)2. If f’(x) < 0 then f is decreasing on (a, b)3. If f’(x) = 0 then f is constant on (a, b)

Definition of Critical Number

If f is defined at c, then c is a critical number of f if f’(c)=0 or if f’ is undefined at c.

Find the open intervals on which the given function is increasing or decreasing.

1. Find derivative.

2. Set f’(x) = 0 and solve to find the critical numbers.

CRITICAL NUMBERS

3. Make table to test the sign f’(x) in each interval.

4. Use the test for increasing/decreasing to decide whether f is increasing or decreasing on each interval.

Interval -∞ < x < -2 -2 < x < 2 2 < x < ∞

Test value x = -3 x = 0 x = 3

Sign of f’(x) f’(-3) > 0 f’(0) < 0 f’(3) > 0

Conclusion Increasing Decreasing Increasing

Find the open intervals on which the given function is increasing or decreasing.

y

x

Incr

easin

g Decreasing

Incr

easin

gRelative maximum

Relative minimum

Definition of Relative Extrema

Let f be a function defined at c.

1. f(c) is a relative maximum of f if there exists an interval (a, b) containing c such that f(x) ≤ f(c) for all x in (a, b).

2. f(c) is a relative minimum of f if there exists an interval (a, b) containing c such that f(x) ≥ f(c) for all x in (a, b).

If f(c) is a relative extremum of f, then the relative extremum is said to occur at x = c.

1. f(c) is a relative maximum of f if there exists an interval (a, b) containing c such that f(x) ≤ f(c) for all x in (a, b).

c

relative maximum f(c)

f(x)

f(x)

f(x)

f(x) f(x)f(x)

f(x)

f(x)

2. f(c) is a relative minimum of f if there exists an interval (a, b) containing c such that f(x) ≥ f(c) for all x in (a, b).

relative minimum f(c)

f(x)f(x)

f(x)

f(x)f(x)

f(x)

Occurrence of Relative Extrema

If f has a relative minimum or a relative maximum when x = c, then c is a critical number of f. That is, either f’(c) = 0 or f’(c) is undefined.

First-Derivative Test for Relative ExtremaLet f be continuous on the interval (a, b) in which c is the only critical number.

On the interval (a, b) if1. f’(x) is negative to the left of x = c and positive to the right of x = c, then f(c) is a relative minimum.2. f’(x) is positive to the left of x = c and negative to the right of x = c, then f(c) is a relative maximum.3. f’(x) has the same sign to the left and right of x = c, then f(c) is not a relative extremum.

1. f’(x) is negative to the left of x = c and positive to the right of x = c, then f(c) is a relative minimum.

Relative minimum

f’(x) is positiv

ef’(x) is

negative

c

2. f’(x) is positive to the left of x = c and negative to the right of x = c, then f(c) is a relative maximum.

relative maximum

f’(x) is positiv

e

f’(x) is negative

c

3. f’(x) has the same sign to the left and right of x = c, then f(c) is not a relative extremum.

c

f’(x) is positive

f’(x) is positive

Not a relative extremum

Find all relative extrema of the given function.

Find derivative

Set = 0 to find critical numbers

CRITICAL NUMBERS

(-∞, -1) (-1, 1) (1, ∞)x = -2 x = 0 x = 2

+ - +Increasing Decreasing Increasing

Relative Maximum (-1, 5)

Relative Minimum (1, -3)

Find all relative extrema of the given function.

(-∞, -2) (-2, 0) (0, ∞)x = -3 x = -1 x = 1

+ - +Increasing Decreasing Increasing

Relative max: (-2, 0) Relative min: (0, -2)

Find all relative extrema of the given function.

(0,π/4) (π/4,3π,4) (3π/4,5π/4) (5π/4,7π/4) (7π/4,2π)

x = π/6 x = π/2 x = π x = 3π/2 x = 2π

+ - + - +

Increasing Decreasing Increasing Decreasing Increasing

Relative max:

Relative min:

The graph of f is shown. Sketch a graph of the derivative of f.

The graph of f is shown. Sketch a graph of the derivative of f.

The graph of f is shown. Sketch a graph of the derivative of f.