Section 2.5Critical Numbers – Relative Maximum and Minimum Points

If f ‘ (a) > 0

The graph of f(x) is INCREASING at x = a

The graph of f ' x is POSITIVE at x = a

f(x)

f '(x)x = a

If f ‘ (a) < 0

The graph of f(x) is DECREASING at x = a

f(x)

f '(x)

The graph of f ' x is aNE t GATIVE x = a

x = a

This is the graph of g(x)This is the graph of f(x)

If f ‘ (a) = 0, a maximum or minimum MAY exist.

If f ‘ (a) = 0, a maximum or minimum MAY exist.

This is the graph of g ‘ (x)This is the graph of f ‘ (x)

A change from increasing to decreasing indicates a maximum

Graph of f(x)

Graph of f ' x

A change from decreasing to increasing indicates a minimum

Graph of f ' x

Graph of f x

•If f ‘ (x) > 0 on an interval (a, b), f is increasing on (a, b).

•If f ‘ (x) < 0 on an interval (a, b), f is decreasing on (a, b).

•If f ‘ (c) = 0 or f ‘ (c) does not exist, c is a critical number

•If f ‘ (c) = 0, a relative maximum will exist IF f ‘ (x) changes from positive to negative.

•If f ‘ (c) = 0, a relative minimum will exist IF f ‘ (x) changes from negative to positive.

•A RELATIVE max/min is a high/low point around the area.

•An ABSOLUTE max/min is THE high/low point on an interval.

FACTS ABOUT f ‘ (x) = 0

A. Where are the relative extrema of f?

B. On what interval(s) is f ‘ < 0?

(1, 3)

C. On what interval(s) is f ‘ > 0?

(-1, 1) and (3, 5)

D. Where are the zero(s) of f?

x = 0

This is the graph of f(x) on the interval [-1, 5].

x = -1, x = 1, x = 3, x = 5

A. Where are the relative extrema of f?

B. On what interval(s) is f ‘ < 0?

[-1, 0)

C. On what interval(s) is f ‘ > 0?

(0, 5]

D. On what interval(s) is f “ > 0?

(-1, 1), (3, 5)This is the graph of f ‘ (x) on the interval [-1, 5].

x = -1, x = 0, x = 5

A. Where are the relative extrema of f?

B. On what interval(s) is f ‘ constant?

(-10, 0)

C. On what interval(s) is f ‘ > 0?

D. For what value(s) of x is f ‘ undefined?

x = -10, x = 0, x = 3

x = -10, x = 3

10, 0 , 0,3

This is the graph of f(x) on [-10, 3].

A. Where are the relative extrema of f?

B. On what interval(s) is f ‘ constant?

none

C. On what interval(s) is f ‘ > 0?

D. For what value(s) of x is f ‘ undefined?

none

x = -10, x = -1, x = 3

1, 3

This is the graph of f ‘ (x) on [-10, 3].

Based upon the graph of f ‘ (x) given f x cos x 1 sinx

on the interval [0, 2pi], answer the following:

Where does f achieve a minimum value? Round your answer(s) to three decimal places.

x = 3.665, x = 6.283

Where does f achieve a maximum value? Round your answer(s)to three decimal places.

x = 0, x = 5.760

CALCULATOR REQUIRED

Estimate to one decimal place the critical numbers of f.

Estimate to one decimal place the value(s) of x at which there is a relative maximum.

-1.4, 0.4

Given the graph of f(x) on to the right, answer the two questions below.

,

-1.4, -0.4, 0.4, 1.6

Estimate to one decimal place the critical numbers of f.

Estimate to one decimal place the value(s) of x at which there is a relative maximum.

1.1

Given the graph of f ‘ (x) on to the right, answer the three questions below.

,

-1.9, 1.1, 1.8

Estimate to one decimal place the value(s) of x at which there is a relative minimum.

-1.9, 1.8

3

2

x 1Given f ' x

x 3

a) For what value(s) of x will f have a horizontal tangent?

1

b) On what interval(s) will f be increasing?

1,

c) For what value(s) of x will f have a relative minimum?

1

d) For what value(s) of x will f have a relative maximum?

none

CALCULATOR REQUIRED

For what value(s) of x is f ‘ (x) = 0? 

 On what interval(s) is f increasing?

. Where are the relative maxima of f?

-1 and 2

x = -1, x = 4

(-3, -1), (2, 4)

This is the graph of f(x) on [-3, 4].

For what value(s) of x if f ‘ (x) = 0? 

For what value(s) of x does a relative maximum of f exist? 

On what interval(s) is f increasing? 

On what interval(s) is f concave up?

-2, 1 and 3

-3, 1, 4

(-2, 1), (3, 4]

(-3, -1), (2, 4)This is the graph of f ‘ (x)

[-3, 4]

This is the graph of f(x) on [-5, 3]

For what values of x if f undefined? 

On what interval(s) is f increasing?

 

On what interval(s) is f ‘ < 0?

 

Find the maximum value of f.

6

(-5, 1)

(1, 3)

-5, 1, 3

This is the graph of f ‘ (x) on [-7, 7].

For what value(s) of x is f ‘ (x) undefined? For what values of x is f ‘ > 0?

 On what interval(s) is f decreasing? 

On what interval(s) is f concave up?

(0, 7)

(-7, 0)

(0, 7]

none

This is the graph of f(x) on [-2, 2].

For what value(s) of x is f ‘ (x) = 0? 

For what value(s) of x does a relative minimum exist? 

On what interval(s) is f ‘ > 0? 

On what interval(s) is f “ > 0?

(-1, 0), (1, 2)

(-2, -1.5), (-0.5, 0.5), (1.5, 2)

-2, -0.5, 1.5

-1.5, -0.5, 0.5, 1.5

This is the graph of f ‘ (x) on [-2, 2]

For what value(s) of x is f ‘ (x) = 0? 

For what value(s) of x is there alocal minimum? 

On what interval(s) is f ‘ > 0? 

On what interval(s) is f “ > 0?

(-2, -1.5), (-0.5, 0.5), (1.5, 2)

(-2, -1), (0, 1)

-2, 0, 2

-2, -1, 0, 1, 2