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Increasing & Decreasing
Function Intervals

Learn how the first derivative reveals where a function rises and falls: master the test that connects \( f'(x) \) to the behavior of \( f(x) \).

AP Calculus AB

What Does "Increasing" Mean?

A function \( f \) is increasing on an interval \( (a, b) \) if for every pair \( x_1 < x_2 \) in that interval, \( f(x_1) < f(x_2) \). In plain language: as you move right, the graph moves up.

x y \(x_1\) \(x_2\) \(f(x_1)\) \(f(x_2)\) \(x_1 < x_2 \implies f(x_1) < f(x_2)\)

What Does "Decreasing" Mean?

A function \( f \) is decreasing on an interval \( (a, b) \) if for every pair \( x_1 < x_2 \) in that interval, \( f(x_1) > f(x_2) \). As you move right, the graph moves down.

x y \(x_1\) \(x_2\) \(f(x_1)\) \(f(x_2)\) \(x_1 < x_2 \implies f(x_1) > f(x_2)\)

The Connection to \( f'(x) \)

The derivative \( f'(x) \) measures the instantaneous rate of change. Its sign tells you the direction the function is heading at every 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: potential local max, min, or inflection.

The First Derivative Test for Intervals

x \(f(x)\) \(f'(x)=0\) \(f'(x)>0\) Increasing \(f'(x)<0\) Decreasing

Critical Numbers: Where Behavior Changes

A critical number \( c \) in the domain of \( f \) satisfies \( f'(c) = 0 \) or \( f'(c) \) is undefined. These are the only points where the function can switch from increasing to decreasing or vice versa.

\( f'(c) = 0 \)
Horizontal tangent: smooth peak or valley (e.g., at the vertex of a parabola).
\( f'(c) \) undefined
Cusp, corner, or vertical tangent: sharp change in direction (e.g., \( f(x) = |x| \) at \( x=0 \)).

Increasing/Decreasing vs. Positive/Negative

Students often confuse the behavior of \( f(x) \) with its sign. A function can be increasing while still being negative.

Increasing / Decreasing Positive / Negative
Asks Is \( f \) going up or down? Is \( f \) above or below the x-axis?
Determined by Sign of \( f'(x) \) Sign of \( f(x) \) itself
Example \( f(x) = x^3 \) at \( x = -1 \): increasing \( f(-1) = -1 \): negative

The Sign Chart Method

1
Find \( f'(x) \). Differentiate the original function.
2
Set \( f'(x) = 0 \). Solve for all critical numbers. Also find where \( f'(x) \) is undefined.
3
Build a sign chart. Plot critical numbers on a number line to divide the domain into intervals.
4
Test one point from each interval in \( f'(x) \). Record the sign: \( + \) or \( - \).
5
State intervals. Where \( f'(x) > 0 \), \( f \) is increasing. Where \( f'(x) < 0 \), \( f \) is decreasing.

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

-1 1 + − + Increasing Decreasing Increasing ◀ ▶ \(f'(x)\) sign chart

Graph of \( f(x) = x^3 - 3x \)

x y -1 1 -2 2 2 -2 local max (−1, 2) local min (1, −2) INC DEC INC

Example 2: \( g(x) = x^4 - 4x^3 \)

0 3 − − + Decreasing Decreasing Increasing ◀ ▶ \(g'(x)\) sign chart

Example 3: \( h(x) = \sin(x) \) on \([0, 2\pi]\)

x y 0 \(\frac{\pi}{2}\) \(\pi\) \(\frac{3\pi}{2}\) \(2\pi\) INC DEC INC

Why Use Open Intervals?

The AP exam and most textbooks state increasing/decreasing intervals using open intervals or closed intervals. Here is the key nuance.

Open Intervals
We test \( f'(x) \) between critical numbers. Since \( f'(c) = 0 \) at the boundary, the derivative is not strictly positive or negative there.
Closed Intervals OK?
If \( f \) is continuous at the endpoint, many textbooks accept closed brackets. The AP exam accepts either convention, but be consistent.

Summary: The Full Process

Compute \(f'(x)\)
Solve \(f'(x) = 0\) and \(f'(x)\) undefined
Build sign chart with test points
Read off + (INC) and − (DEC)
State intervals in answer

Common Mistakes to Avoid

✕ Checking \( f(x) \) instead of \( f'(x) \)
Students plug test points into the original function instead of the derivative. You need the sign of \( f'(x) \), not \( f(x) \).
✕ Assuming \( f'(c) = 0 \) means extremum
A zero derivative does not guarantee a max or min. Example: \( f(x) = x^3 \) has \( f'(0) = 0 \) but no extremum at \( x = 0 \).
✕ Forgetting undefined points
If \( f'(x) \) has a denominator, points where it is undefined are also critical. Example: \( f(x) = x^{2/3} \) has \( f'(0) \) undefined.

Question 1 of 5

Given \( f(x) = 2x^3 - 9x^2 + 12x \), on which interval is \( f \) increasing?

Question 2 of 5

If \( f'(x) > 0 \) for all \( x \) in \( (2, 5) \), which statement is true?

Question 3 of 5

Let \( g(x) = x^4 - 8x^2 \). On which interval is \( g \) decreasing?

Question 4 of 5

The function \( f(x) = xe^{-x} \) has a critical number at \( x = 1 \). What is the behavior of \( f \) around this point?

Question 5 of 5

If \( f'(c) = 0 \) but \( f'(x) < 0 \) on both sides of \( c \), what can you conclude?