Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Examples and instructions on how to determine the intervals where a given function is increasing or decreasing. It includes the function's formula, examples, and test for increasing and decreasing functions. Students of calculus and related fields will find this document useful for understanding the concepts of increasing and decreasing functions.
Typology: Study notes
1 / 6
This page cannot be seen from the preview
Don't miss anything!
Section 10.1 A
Increasing and Decreasing Functions
8
6
4
2
-10 -5 5 10
Example 1
Give the intervals where the function is increasing and decreasing.
8
6
4
2
-10 -5 5 10
f x( ) = x^2 -3⋅x
Decreasing: (^)
Increasing: (^)
Give the intervals where the function increasing and decreasing
8
6
4
2
-10 -5 5 10
f x( ) = x^3 -3⋅x
Increasing: ( −∞, − 1 )∪( 1 ,∞) Decreasing: (− 1 , 1 )
Test for increasing and decreasing functions
Let f be a differentiable function on the interval
If f ′( x )> 0 for all x in (a,b), then f is increasing on (a,b)
If f ′(^ x )< 0 for all x in (a,b), then f is decreasing on (a,b)
If f ′(^ x )= 0 for all x in (a,b), then f is constant on (a,b)
Find the intervals where the function is decreasing and increasing
3 f x = x −
2
2
2
2
2
2
f
f
x
x
x
x
f x x
Interval (−∞, 0 ) ( 0 ,∞)
Test Value x =− 1 x = 1
Sign of f ′(^ x ) Positive^ Positive
Conclusion Increasing Increasing
Find the intervals where the function is increasing or decreasing.
3 2 f ( x )= x − 3 x
2
2
x x
orx
x
x orx
xx
x x
f x x x
2
2
2
f
f
f
Interval (−∞, 0 ) ( 0 , 2 ) ( 2 ,∞)
Test Value x =− 1 x = 1 x = 3
Sign of f ′(^ x ) Positive^ Negative^ Positive
Conclusion Increasing Decreasing Increasing
