Increasing or Decreasing Intervals

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Increasing/Decreasing Test Theorem
let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b).
1. if f '(x) > 0 for all x in (a, b), then f is increasing on [a, b].
2. if f '(x) < 0 for all x in (a, b), then f is decreasing on [a, b].
3. if f '(x) = 0 for all x in (a, b), then f is constant on [a, b].

Finding Intervals where a Function is Increasing or Decreasing

1.	Locate the critical numbers of f in (a, b), and use these numbers to determine test intervals.
2.	Determine the sign of f '(x) at one value in each of the test intervals.
3.	Use the Increasing/Decreasing Test Theorem to decide whether f is increasing or decreasing on each interval.

Example:

Find the open intervals on which f(x) = x³ - x² is increasing or decreasing.

Get the Critical Numbers{ see Finding Minimum and Maximum Extrema}
x= 0, 2/3 see table for test intervals

Determine the sign of f '(x)
see table below

Use the Increasing/Decreasing Test Theorem
see table below

Interval	-infinity < x < 0	0 < x < 2/3	2/3 < x < infinity
Test Value	-1	1/3	3
Sign of f '(x)	f '(-1) = 5 > 0	f '(1/3) = -1/3 < 0	f '(3) = 21 >0
Theorem	Increasing	Decreasing	Increasing