Section 4.3First and Second

Derivative Information

Test for Increasing or Decreasing Functions

Let f be continuous on [a,b] and differentiable on (a,b).

1. If ( ) 0 for all x in (a,b), then f is increasing on [a,b].

2. If ( ) 0 for all x in (a,b), then f is decreasing on [a,b].

3. If ( ) 0 for all x in (a,b), then f is constant on [a,b].

f x

f x

f x

Increasing/Decreasing

To determine whether the function is increasing or decreasing on an interval, evaluate points to the left and right of the critical points on an f ’ numberline.

f'(x) c

+ __inc

__dec

f'(x) c

+__incdec

First Derivative Test

Let c be a critical number of a function f that is continuous on an open interval containing c. If f is differentiable on the interval, except possibly at c, then f(c) can be classified as follows…

1. If ( ) changes from negative to positive at c,

then f(c) is a relative minimum.

2. If ( ) changes from positive to negative at c,

then f(c) is a relative maximum.

f x

f x

1st Derivative Test

f'(x) c

+ __inc

__dec

If the sign changes from + to - at c, then c is a relative maximum.

Max

f'(x) c

+__incdec

If the sign changes from - to + at c, then c is a relative minimum.

Min

• A curve is concave up if its slope is increasing, in which case the second derivative will be positive ( f "(x) >0 ).

• Also, the graph lies above its tangent lines.

•A curve is concave down if its slope is decreasing, in which case the second derivative will be negative (f "(x) < 0 ).• Also, the graph lies below its tangent lines.

Concavity

Test for Concavity

Let f be a function whose 2nd derivative exists on an open interval I.

1. If ( ) 0 for all x in I, then f is concave upward.

2. If ( ) 0 for all x in I, then f is concave downward.

f x

f x

Concavity Test

To determine whether a function is concave up or concave down on an interval, determine where f "(x) = 0 and f "(x) is undefined. Then evaluate values to the left and right of these points on an f " numberline.

f "(x)c

+ __ccu

__ccd

f "(x)c

+__ccuccd

A point where the graph of f changes concavity, from concave up to concave down or vice versa, is called a point of inflection. At a point of inflection the second derivative will either be undefined or 0.

Inflection

If the signs on the f "(x) numberline do not change, then c is not an inflection point.

f "(x) c

__ccdccd __

Not inflection

When the signs change on an f " numberline, there is an inflection point.

Second Derivative Test

Let f be a function such that f ’(c) = 0 and the 2nd derivative of f exists on an open interval containing c.

1. If ( ) 0 , then f(c) is a relative minimum.

2. If ( ) 0, then f(c) is a relative maximum.

3. If ( ) 0, then the test fails. Use the 1st Derivative Test.

f c

f c

f c