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Increasing & Decreasing
Function Intervals

Learn how to use the first derivative to determine where a function rises and falls: the foundation for curve sketching and optimization.

AP Calculus BC

What Does Increasing and Decreasing Mean?

Increasing
A function \( f \) is increasing on an interval if, for any \( a < b \) in that interval, \( f(a) < f(b) \). The graph moves upward from left to right.
Decreasing
A function \( f \) is decreasing on an interval if, for any \( a < b \) in that interval, \( f(a) > f(b) \). The graph moves downward from left to right.

Visual Intuition: Reading the Graph

x y local max local min increasing decreasing increasing

The Derivative Connection

The sign of the first derivative tells us everything about whether a function is increasing or decreasing. The derivative \( f'(x) \) measures the instantaneous rate of change: it is the slope of the tangent line at each point.

\( f'(x) > 0 \)
Tangent line slopes upward. The function is increasing.
\( f'(x) < 0 \)
Tangent line slopes downward. The function is decreasing.
\( f'(x) = 0 \)
Tangent line is horizontal. Possible local max, min, or inflection.

The Increasing/Decreasing Test

This theorem converts a question about the original function into a question about the sign of its derivative: a much simpler algebraic problem.

Critical Points: Where the Sign Can Change

A critical number of \( f \) is a value \( c \) in the domain of \( f \) where either:

1
\( f'(c) = 0 \): the derivative equals zero (horizontal tangent)
2
\( f'(c) \) does not exist: the derivative is undefined (corner, cusp, or vertical tangent)

Critical Point vs. Extremum

Critical Point Local Extremum
Definition \( f'(c) = 0 \) or DNE Local max or min value
Sign change required? No Yes: \( f' \) must change sign
Example \( f(x) = x^3 \) at \( x=0 \) \( f(x) = x^2 \) at \( x=0 \)
Key insight Every extremum is a critical point Not every critical point is an extremum

How to Find the Intervals

1
Differentiate. Find \( f'(x) \).
2
Find critical numbers. Solve \( f'(x) = 0 \) and identify where \( f'(x) \) is undefined.
3
Build a sign chart. Place critical numbers on a number line, creating test intervals.
4
Test each interval. Pick a sample point in each interval and evaluate the sign of \( f'(x) \).
5
State intervals. Positive \( \Rightarrow \) increasing. Negative \( \Rightarrow \) decreasing. Write in interval notation.

Example 1: \( f(x) = x^3 - 3x + 1 \)

1
Differentiate: \( f'(x) = 3x^2 - 3 = 3(x^2 - 1) = 3(x-1)(x+1) \)
2
Critical numbers: \( f'(x) = 0 \) when \( x = -1 \) and \( x = 1 \)
3
Test intervals: \( (-\infty, -1) \), \( (-1, 1) \), \( (1, \infty) \)
-1 1
\( f' > 0 \)
\( f' < 0 \)
\( f' > 0 \)
 INC DEC INC

Example 1: Graph Confirmation

x y -1 1 (-1, 3) (1, -1) 3 1 -1

Example 2: \( g(x) = x \, e^{-x} \)

1
Differentiate using the product rule: \( g'(x) = e^{-x} + x(-e^{-x}) = e^{-x}(1-x) \)
2
Critical number: Since \( e^{-x} > 0 \) always, set \( 1-x=0 \): \( x=1 \)
3
Test: For \( x < 1 \), \( 1-x > 0 \), so \( g'>0 \). For \( x > 1 \), \( 1-x < 0 \), so \( g'<0 \).

Example 3: \( h(x) = x^{2/3}(x-5) \)

1
Rewrite: \( h(x)=x^{5/3}-5x^{2/3} \). Then \( h'(x)=\frac{5}{3}x^{2/3}-\frac{10}{3}x^{-1/3}=\frac{5x-10}{3x^{1/3}} \).
2
Critical numbers: \( h'(x)=0 \) when \( 5x-10=0 \Rightarrow x=2 \). Also \( h'(x) \) is undefined at \( x=0 \).
3
Three intervals: \( (-\infty,0) \), \( (0,2) \), \( (2,\infty) \)
0 2
\( h' > 0 \)
\( h' < 0 \)
\( h' > 0 \)
 INC DEC INC

Why the Sign Chart Works

The sign chart method relies on a powerful theorem from real analysis:

This means we only need to test one point per interval. If \( f' \) is positive at that test point, it is positive throughout the entire interval.

Open vs Closed Interval Notation

There is a subtle convention difference in how textbooks state increasing/decreasing intervals. Both are accepted on the AP exam:

Open intervals
\( f \) is increasing on \( (-1, 2) \). This is where \( f' > 0 \) strictly. Most common on AP.
Closed intervals
\( f \) is increasing on \( [-1, 2] \). This includes the endpoints where continuity guarantees the behavior extends.

Common Mistakes to Avoid

Using \( f \) instead of \( f' \)
You test the sign of the derivative \( f'(x) \), not the original function \( f(x) \). The sign of \( f \) tells you nothing about increasing/decreasing.
Forgetting DNE critical points
If \( f'(c) \) is undefined but \( c \) is in the domain of \( f \), it is still a critical point. Cusps and corners count!
Using x-values as intervals
Write intervals, not points. Say "increasing on \( (1, 4) \)," not "increasing at \( x = 2 \)."

Summary: The Complete Picture

Sign of \( f'(x) \) Behavior of \( f \) Graph Direction
\( f'(x) > 0 \) Increasing Rising left to right
\( f'(x) < 0 \) Decreasing Falling left to right
\( f'(x) = 0 \) Critical point Horizontal tangent
\( f'(x) \) DNE Critical point Corner, cusp, or vertical tangent
\( f' \) changes + to - Local maximum Peak
\( f' \) changes - to + Local minimum Valley

Question 1 of 5

If \( f'(x) = (x-2)(x+3) \), on which interval is \( f \) decreasing?

Question 2 of 5

Given \( f(x) = \dfrac{x^2}{x^2+4} \), find where \( f \) is increasing.

Question 3 of 5

The function \( f \) has derivative \( f'(x) = e^x(x^2 - 1) \). At how many points does \( f \) have a local extremum?

Question 4 of 5

If \( f'(x) = x^2(x-4) \), which statement about \( f \) is correct?

Question 5 of 5

Let \( f(x) = \sin(x) + \cos(x) \) on \( [0, 2\pi] \). On which interval is \( f \) increasing?