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
Domain too ,oo )since (6+3) is
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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
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