12 Critical Numbers
of 6
Transcript of 12 Critical Numbers
-
8/8/2019 12 Critical Numbers
1/6
Mr. Simonds MTH 251 - Critical Numbers - Increasing/Decreasing Concavity LMPs, LmPs, and IPs
Pay attention to the writing guidelines as well as the math - Page 1 of 6
Example 1Consider the function f shown in Figure 1.
a. What are the critical numbers of f ? Explain.
b. Where does f change concavity? What else happens at these points?
c. What are the inflection points on f ? Local max points? Local min points?
The critical numbers of f are the values of x in the domain of f where either
( ) 0 f x = or ( ) f x is undefined. These numbers are important because they are the onlyvalues of x where f could possibly have local extreme values.
Figure 1
y
-
8/8/2019 12 Critical Numbers
2/6
Mr. Simonds MTH 251 - Critical Numbers - Increasing/Decreasing Concavity LMPs, LmPs, and IPs
Pay attention to the writing guidelines as well as the math - Page 2 of 6
Example 2
Find the critical numbers of ( )( )
2
1/3
lng
= .
Example 3
Find the critical numbers of ( ) ( ) ( )34 34 += x x x f .
-
8/8/2019 12 Critical Numbers
3/6
Mr. Simonds MTH 251 - Critical Numbers - Increasing/Decreasing Concavity LMPs, LmPs, and IPs
Pay attention to the writing guidelines as well as the math - Page 3 of 6
Example 4
The first derivative of the function ( )3
2
10 xT x
x
= is ( ) ( )
( )23 35 12
3 10
xT x
x x
=
.
a. What are the critical numbers of T ? Explain.
b. Build a table that clearly shows where T is increasing and where T is decreasing. Make sure your table communicates how you arrived at your conclusions.
c. What are the local maximum points on T ? Local minimum points? What sort of point is( )10,0 ?
-
8/8/2019 12 Critical Numbers
4/6
Mr. Simonds MTH 251 - Critical Numbers - Increasing/Decreasing Concavity LMPs, LmPs, and IPs
Pay attention to the writing guidelines as well as the math - Page 4 of 6
Example 5 Find the local minimum points, local maximum points, and inflection points on the function
( ) 3 31
h x x x
= + . Show all relevant work.
-
8/8/2019 12 Critical Numbers
5/6
Mr. Simonds MTH 251 - Critical Numbers - Increasing/Decreasing Concavity LMPs, LmPs, and IPs
Pay attention to the writing guidelines as well as the math - Page 5 of 6
Example 6
Determine the intervals over which the function ( ) ( ) ( ) ( )
= sincos2
cos 2
f is
increasing and the intervals over which the function is decreasing.
-
8/8/2019 12 Critical Numbers
6/6
Mr. Simonds MTH 251 - Critical Numbers - Increasing/Decreasing Concavity LMPs, LmPs, and IPs
Pay attention to the writing guidelines as well as the math - Page 6 of 6
Example 7
Find the critical numbers of ( )4
10
t k t
t
=
.
