Chapter 4, Section 4.3: Derivatives and Shapes of the Curves

Increasing/Decreasing Test:

(a) If f 0(x) > 0 on an interval, then f is increasing on that interval

(b) If f 0(x) < 0 on an interval, then f is decreasing on that interval

Example 1: Find the intervals on which f(x) =1

4x

4 � 23x

3 is increasing and decreasing.

The First Derivative Test: Suppose that c is a critical number of a continuous function f .

(a) If f 0(x) changes from positive to negative around c, then f has a local maximum at c.

(b) If f 0(x) changes from negative to positive around c, then f has a local minimum at c.

(c) If f 0(x) does not change sign around c (for example, f 0(x) is positive on both sides of c or negative

on both sides), then f has no local maximum or minimum at c.

Example 2: Find the local maximum and local minimum for f(x) = x8(x� 2)7

Chapter 4, Sec4.3: Derivatives and Shapes of the Curves

Concavity Test:

(a) If f 00(x) > 0 for all x in I, then the graph of f is concave up on I.

(b) If f 00(x) < 0 for all x in I, then the graph of f is concave down on I.

(c) A point where a curve changes it’s direction of concavity is called an inflection point .

Example 3: Let f(x) = x3 ln x. Find the intervals where f(x) is concave up and concave down and

find any inflection points. (Round all answers to three decimal places)

Example 4: Suppose the derivative of a continuous function f is given below. On what interval is f

increasing?

f

0(x) = (x+ 2)4(x� 5)5(x� 6)6

2 Spring 2017, c�Maya Johnson

Chapter 4, Sec4.3: Derivatives and Shapes of the Curves

Example 5: The graph of the first derivative f 0 of a function f is shown below, find

(a) On what interval(s) is f increasing/decreasing?

(b) At what value(s) of x does f have local maximum?

(c) At what value(s) of x does f have local minimum?

(d) On what interval(s) is f concave upward?

(e) On what interval(s) is f concave downward?

(f) What are the x�coordinate(s) of the inflection point(s) of f?

3 Spring 2017, c�Maya Johnson

Chapter 4, Sec4.3: Derivatives and Shapes of the Curves

Example 6: Let f(x) = e7x + e�x.

(a) On what interval(s) is f increasing/decreasing?

(b) At what value(s) of x does f have a local maximum?

(c) At what value(s) of x does f have a local minimum?

(d) On what interval(s) is f concave upward?

(e) On what interval(s) is f concave downward?

(f) What are the x�coordinate(s) of the inflection point(s) of f?

4 Spring 2017, c�Maya Johnson

Chapter 4, Sec4.3: Derivatives and Shapes of the Curves

Example 7: Let f(x) =x

2

x

2 + 3.

(a) On what interval(s) is f increasing/decreasing?

(b) At what value(s) of x does f have a local maximum?

(c) At what value(s) of x does f have a local minimum?

(d) On what interval(s) is f concave upward?

(e) On what interval(s) is f concave downward?

(f) What are the x�coordinate(s) of the inflection point(s) of f?

5 Spring 2017, c�Maya Johnson

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Chapter 4, Sec4.3: Derivatives and Shapes of the Curves

Example 8: Given f(x) = 3x2/3 � x.

(a) On what interval(s) is f increasing/decreasing?

(b) At what value(s) of x does f have a local maximum?

(c) At what value(s) of x does f have a local minimum?

(d) On what interval(s) is f concave upward?

(e) On what interval(s) is f concave downward?

(f) What are the x�coordinate(s) of the inflection point(s) of f?

6 Spring 2017, c�Maya Johnson

Domain too , A )

f 'hn=2x"' 3-1=0

Chapter 4, Sec4.3: Derivatives and Shapes of the Curves

The Second Derivative Test: Suppose f 00 is continuous near c.

(a) If f 0(c) = 0 and f 00(c) > 0, then f has local minimum at c.

(b) If f 0(c) = 0 and f 00(c) < 0, then f has local maximum at c.

Example 9 Find the critical numbers of the function and describe the behavior of f at these numbers.

f(x) = x10(x� 4)9

Example 10: Assuming that the function f(x) is continuous on the interval (�1,1), indicate whethereach of the points listed below is a relative maximum, relative minimum, neither or cannot be determined

from the information given.

(a) (1, f(1)) if f 0(1) = 0 and f 00(1) < 0

(b) (0, f(0)) if f 0(0) = �3 and f 00(0) < 0

(c) (�1, f(�1)) if f 0(�1) = 0 and f 00(1) > 0

(d) (3, f(3)) if f 0(3) = 0 and f 00(3) = 0

7 Spring 2017, c�Maya Johnson