In this page increasing and decreasing intervals we are going to discuss about how to find increasing and decreasinginterval for any function.
Procedure to find where the function is increasing or decreasing:
Example 1:
Find the intervals in which f (x) = 2x³ + x²  20 x is increasing or decreasing
Solution:
f (x) = 2x³ + x²  20 x
f '(x) = 2(3x²) + 2 x  20
f '(x) = 6x² + 2 x  20
÷ by 2 ⇒ 3x² + x  10
f '(x) = 0
3x² + x  10 = 0
(3x  5) (x + 2) = 0
( 3x  5) = 0 ( x + 2) = 0
3 x = 5 x =  2
x = 5/3
We can split this into three intervals (∞,2) (2,5/3) (5/3,∞). Now let us see the given function is increasing or decreasing in which intervals.
Interval  (3x5)  (x+2)  f ' (x)  Intervals of increasing/decreasing 

∞ < x < 2  

 
2 < x < 5/3  

 
5/3 < x < ∞  


The given is increasing on (∞,2] ∪ [5/3,∞) and decreasing on [2,5/3]
Example 2:
Find the intervals in which f (x) = x³  3 x + 1 is increasing or decreasing
Solution:
f (x) = x³  3 x + 1
f '(x) = (3x²)  3 (1)
f '(x) = 3x²  3
÷ by 3 ⇒ x²  1
f '(x) = 0
x²  1 = 0
(x + 1) (x  1) = 0
(x + 1) = 0 (x  1) = 0
x = 1 x = 1
We can split this as three intervals (∞,1) (1,1) (1,∞). Now let us see the given function is increasing or decreasing in which intervals.
Interval  (x+1)  (x1)  f ' (x)  Intervals of increasing/decreasing 

∞ < x < 1  

 
1 < x < 1  

 
1 < x < ∞  


The given is increasing on (∞,1] ∪ [1,∞) and decreasing on [1,1]
Example 3:
Find the intervals in which f (x) = x  2 sin x is increasing or decreasing
Solution:
f (x) = x  2 sin x
f '(x) = 1  2 cos x
f '(x) = 0
1  2 cos x = 0
 2 cos x = 1
cos x = 1/2
x = cos ⁻ ¹ (1/2)
x = Π/3,5Π/3
x = Π/3 x = 5Π/3
increasing and decreasing intervals increasing and decreasing intervals
We can split this as three intervals (0,Π/3) (Π/3,5Π/3) (5Π/3,2Π). Now let us see the given function is increasing or decreasing in which intervals.
Interval  1  2 cos x  f ' (x)  Intervals of increasing/decreasing 

0 < x < Π/3  


Π/3 < x < 5Π/3  
 
5Π/3 < x < 2Π  

The given is increasing on [Π/3,5,Π/3] and decreasing on (0,Π/3]
∪ [5Π/3,2Π).
These are the examples in the topic increasing and decreasing intervals. By practicing these kinds of problems you can understand this topic clearly.
Related Topics
Quote on Mathematics
“Mathematics, without this we can do nothing in our life. Each and everything around us is math.
Math is not only solving problems and finding solutions and it is also doing many things in our day to day life. They are:
It subtracts sadness and adds happiness in our life.
It divides sorrow and multiplies forgiveness and love.
Some people would not be able accept that the subject Math is easy to understand. That is because; they are unable to realize how the life is complicated. The problems in the subject Math are easier to solve than the problems in our real life. When we people are able to solve all the problems in the complicated life, why can we not solve the simple math problems?
Many people think that the subject math is always complicated and it exists to make things from simple to complicate. But the real existence of the subject math is to make things from complicate to simple.”