Determining Intervals on Which a Function Is Increasing or Decreasing

students, one of the most useful ideas in AP Calculus AB is learning how the graph of a function behaves without drawing every tiny detail by hand 📈. If you can tell where a function is increasing or decreasing, you can understand its shape, predict turning points, and solve real-world problems like maximizing profit or modeling motion.

Learning objectives:

Explain what it means for a function to be increasing or decreasing.
Use the derivative to determine intervals where a function rises or falls.
Connect critical points to changes in increasing and decreasing behavior.
Apply these ideas to graph analysis and optimization.
Use AP Calculus AB reasoning to support conclusions with evidence.

A big idea in this lesson is simple: the sign of the derivative tells you how the function is moving. When the derivative is positive, the function is increasing. When the derivative is negative, the function is decreasing. That connection is one of the foundation stones of analytical applications of differentiation.

What It Means for a Function to Increase or Decrease

A function is increasing on an interval when larger input values give larger output values. In everyday language, the graph moves upward as you read from left to right. A function is decreasing on an interval when larger input values give smaller output values, so the graph moves downward from left to right.

Suppose $f$ is a function. On an interval $I$:

$f$ is increasing if for any $a$ and $b$ in $I$, whenever $a<b$, then $f(a)<f(b)$.
$f$ is decreasing if for any $a$ and $b$ in $I$, whenever $a<b$, then $f(a)>f(b)$.

This definition is about comparing two points. On a graph, though, students often think only about whether the line “goes up” or “goes down.” That picture is helpful, but AP Calculus wants the reasoning behind the picture too. The derivative provides that reason.

For a function that is differentiable on an interval, the derivative $f'(x)$ measures the instantaneous rate of change. If $f'(x)>0$, the slope of the tangent line is positive, so the graph is rising. If $f'(x)<0$, the slope is negative, so the graph is falling. This is the calculus version of increasing and decreasing behavior.

How the Derivative Reveals Behavior

The most important test for increasing and decreasing is based on the sign of $f'(x)$.

If $f'(x)>0$ on an interval, then $f$ is increasing on that interval.
If $f'(x)<0$ on an interval, then $f$ is decreasing on that interval.
If $f'(x)=0$ at a point, that point may be important, but it does not by itself tell the whole story.

Think of $f'(x)$ as a speedometer for the graph. Positive derivative means the function is “moving uphill.” Negative derivative means it is “moving downhill.” A derivative equal to $0$ means the graph is momentarily flat, but it might still be changing from increasing to decreasing or from decreasing to increasing.

Here is a simple example.

Let $f(x)=x^2$. Then $f'(x)=2x$.

For $x<0$, $f'(x)<0$, so $f$ is decreasing on $(-\infty,0)$.
For $x>0$, $f'(x)>0$, so $f$ is increasing on $(0,\infty)$.

This matches the familiar parabola shape. The function goes down as $x$ moves toward $0$ from the left, then goes up after $0$. The point $x=0$ is a turning point and also a critical point because $f'(0)=0$.

Critical Points and Sign Charts

To determine intervals of increase and decrease, the first step is often to find critical points. A critical point occurs where $f'(x)=0$ or where $f'(x)$ does not exist, as long as $f$ itself is defined there.

Why are critical points important? Because the sign of $f'(x)$ can change only at points where the derivative is $0$ or undefined. So these points break the number line into intervals to test.

A common AP method is to make a sign chart:

Find $f'(x)$.
Set $f'(x)=0$ and solve.
Identify values where $f'(x)$ is undefined.
Use those values to split the real line into intervals.
Test the sign of $f'(x)$ on each interval.
Conclude where $f$ is increasing or decreasing.

Example: Let $f(x)=x^3-3x$.

First compute the derivative:

$$f'(x)=3x^2-3=3(x^2-1)=3(x-1)(x+1)$$

Set $f'(x)=0$:

$$3(x-1)(x+1)=0$$

so the critical numbers are $x=-1$ and $x=1$.

Now test intervals:

On $(-\infty,-1)$, choose $x=-2$. Then $f'(-2)=9>0$.
On $(-1,1)$, choose $x=0$. Then $f'(0)=-3<0$.
On $(1,\infty)$, choose $x=2$. Then $f'(2)=9>0$.

So $f$ is increasing on $(-\infty,-1)$ and $(1,\infty)$, and decreasing on $(-1,1)$. This is a classic pattern: increasing, then decreasing, then increasing again.

Connecting Increasing and Decreasing to the Graph

The derivative does more than tell us where a function rises or falls. It also helps us analyze the graph itself.

If a function changes from increasing to decreasing at $x=c$, then $f(c)$ is a local maximum. If it changes from decreasing to increasing at $x=c$, then $f(c)$ is a local minimum. These are called turning points because the graph changes direction there.

However, not every critical point is a maximum or minimum. For example, the function $f(x)=x^3$ has derivative $f'(x)=3x^2$, which is never negative. The function is increasing on both sides of $x=0$, even though $f'(0)=0$. So a zero derivative does not automatically mean the function changes direction.

This is why AP Calculus emphasizes the first derivative test. Instead of looking only at $f'(c)=0$, you look at whether $f'(x)$ changes sign around $c$.

If $f'(x)$ changes from positive to negative, then $f$ has a local maximum.
If $f'(x)$ changes from negative to positive, then $f$ has a local minimum.
If there is no sign change, there is no local extremum at that point.

This same method helps with graph analysis. A sketch of $f$ becomes much easier when you know where the function is increasing, decreasing, and turning.

A Real-World Meaning: Motion and Change

Increasing and decreasing intervals are not just abstract graph ideas. They show up in real situations like motion, population growth, and business.

For example, if $s(t)$ gives the position of an object at time $t$, then $s'(t)$ is velocity.

If $s'(t)>0$, the object is moving in the positive direction.
If $s'(t)<0$, the object is moving in the negative direction.

So determining intervals where $s(t)$ is increasing or decreasing tells you when the object is moving forward or backward. A stopped object may have $s'(t)=0$, but that alone does not tell you whether it will turn around next.

In business, if $P(x)$ is profit as a function of the number of items sold, then intervals of increase mean profits are rising as production or sales increase. Intervals of decrease mean profit is falling. That information is useful before doing optimization, since maximum profit often occurs where the function switches from increasing to decreasing.

Common Mistakes to Avoid

students, this topic has a few frequent traps ⚠️.

First, do not confuse the function value with the derivative. A function can be above the $x$-axis and still be decreasing. The sign of $f(x)$ is not the same as the sign of $f'(x)$.

Second, do not assume every critical point is a maximum or minimum. Always check whether $f'(x)$ changes sign.

Third, remember to give intervals using parentheses, not brackets, when discussing increasing or decreasing. Critical points usually separate the intervals, so the endpoints are excluded.

Fourth, if $f'(x)$ is undefined at a point, that point may still be a critical number if $f$ is defined there. You still need to test the intervals around it.

For example, if $f(x)=|x|$, then $f'(x)=-1$ for $x<0$ and $f'(x)=1$ for $x>0$, but $f'(0)$ does not exist. The function is decreasing on $(-\infty,0)$ and increasing on $(0,\infty)$. The sharp corner at $0$ is important even though the derivative is undefined there.

Conclusion

Determining intervals on which a function is increasing or decreasing is a core skill in AP Calculus AB because it connects derivatives to graph behavior. Once you know how to find $f'(x)$, identify critical points, and test signs on intervals, you can explain how a function moves across the coordinate plane. This skill supports graph sketching, local extrema, motion analysis, and optimization. In short, the derivative gives you a powerful way to read the story of a function.

Study Notes

A function is increasing when larger $x$-values produce larger $f(x)$-values.
A function is decreasing when larger $x$-values produce smaller $f(x)$-values.
If $f'(x)>0$ on an interval, then $f$ is increasing there.
If $f'(x)<0$ on an interval, then $f$ is decreasing there.
Critical numbers occur where $f'(x)=0$ or where $f'(x)$ does not exist, provided $f$ is defined there.
Use a sign chart to test the sign of $f'(x)$ on intervals created by critical numbers.
A change in $f'(x)$ from positive to negative indicates a local maximum.
A change in $f'(x)$ from negative to positive indicates a local minimum.
If $f'(x)=0$ but the sign does not change, the point is not necessarily a max or min.
Intervals of increase and decrease help with graph analysis, motion, and optimization.
Always write interval answers with parentheses, such as $(-1,2)$ or $(2,\infty)$.
This topic is a key part of analytical applications of differentiation and connects directly to the first derivative test.