Determining Intervals on Which a Function Is Increasing or Decreasing

students, imagine watching a hill on a bike ride 🚴. Some parts go uphill, some go downhill, and some parts flatten out. In calculus, we use the derivative to describe the same idea for a function. Today’s goal is to understand how to determine where a function is increasing, decreasing, or staying flat, using derivative information.

By the end of this lesson, you should be able to:

explain what it means for a function to be increasing or decreasing,
use the derivative to find intervals of increase and decrease,
connect this skill to graph analysis and other applications of differentiation,
and interpret results in a way that matches AP Calculus BC reasoning.

This topic is a major part of analytical applications of differentiation because it helps us describe the shape and behavior of a function without needing to plot every point. That is a powerful tool in both math and real life 📈.

What Increasing and Decreasing Mean

A function is increasing on an interval if larger input values produce larger output values. In simple language, as $x$ moves to the right, the graph rises. A function is decreasing on an interval if larger input values produce smaller output values, so the graph falls.

For example, if $f(1)=2$, $f(2)=4$, and $f(3)=6$, then the function is increasing over that range because the output gets bigger as $x$ increases. If instead $f(1)=6$, $f(2)=4$, and $f(3)=2$, the function is decreasing.

On a graph, this is easy to picture. But in calculus, we often need to determine these intervals from a formula rather than a picture. That is where the derivative comes in.

The derivative $f'(x)$ tells us the slope of the tangent line to the graph of $f$ at each point. If $f'(x)>0$, the slopes are positive, and the function is increasing. If $f'(x)<0$, the slopes are negative, and the function is decreasing. If $f'(x)=0$, the graph may be flat at that moment, but that does not automatically mean the function changes direction there.

This idea is one of the most important connections in calculus:

$f'(x)>0$ means $f$ is increasing,
$f'(x)<0$ means $f$ is decreasing,
and $f'(x)=0$ may indicate a critical point.

How to Find Intervals of Increase and Decrease

The standard AP Calculus process is to study the sign of $f'(x)$.

Here is the basic procedure:

Find the derivative $f'(x)$.
Find the critical numbers by solving $f'(x)=0$ or identifying where $f'(x)$ does not exist, as long as $f(x)$ is defined there.
Use those critical numbers to split the number line into intervals.
Test the sign of $f'(x)$ on each interval.
State where $f$ is increasing or decreasing.

This is often done using a sign chart or sign analysis.

Suppose $f'(x)=(x-2)(x+1)$. The critical numbers are $x=-1$ and $x=2$. These divide the number line into three intervals: $(-\infty,-1)$, $(-1,2)$, and $(2,\infty)$.

Now test one value from each interval:

For $x=-2$, $f'(-2)=(-4)(-1)>0$.
For $x=0$, $f'(0)=(-2)(1)<0$.
For $x=3$, $f'(3)=(1)(4)>0$.

So the function is increasing on $(-\infty,-1)$ and $(2,\infty)$, and decreasing on $(-1,2)$.

Notice something important: the intervals are written using open intervals because behavior is described between critical numbers, not at the critical numbers themselves.

Why the Sign of the Derivative Matters

The derivative gives the slope of the graph. A positive slope means the graph rises from left to right, and a negative slope means it falls from left to right. This is the calculus version of “uphill” and “downhill” ⛰️.

Think about driving a car on a road. If the road slopes upward, your elevation increases as you move forward. If it slopes downward, your elevation decreases. The derivative is like a slope meter for a function.

This also connects to the Mean Value Theorem. The theorem says that if a function is continuous on $[a,b]$ and differentiable on $(a,b)$, then there is some $c$ in $(a,b)$ such that

$$f'(c)=\frac{f(b)-f(a)}{b-a}.$$

This means the instantaneous slope matches the average slope somewhere in the interval. That helps explain why derivative sign gives information about whether a function is rising or falling overall.

If $f'(x)>0$ on an interval, then every tangent slope there is positive, so the function must increase on that interval. If $f'(x)<0$ on an interval, the function must decrease there.

Example With a Polynomial

Let $f(x)=x^3-3x^2+2$.

First, compute the derivative:

$$f'(x)=3x^2-6x=3x(x-2).$$

Now find critical numbers by setting $f'(x)=0$:

$$3x(x-2)=0,$$

so $x=0$ and $x=2$.

These split the real line into three intervals: $(-\infty,0)$, $(0,2)$, and $(2,\infty)$.

Test one value in each interval:

At $x=-1$, $f'(-1)=3(-1)(-3)>0$.
At $x=1$, $f'(1)=3(1)(-1)<0$.
At $x=3$, $f'(3)=3(3)(1)>0$.

Therefore:

$f$ is increasing on $(-\infty,0)$,
decreasing on $(0,2)$,
increasing on $(2,\infty)$.

This example also shows that a function can increase, then decrease, then increase again. That kind of pattern is common on AP problems and is often connected to local maxima and minima.

Critical Numbers and What They Tell Us

A critical number is an $x$-value in the domain of $f$ where either $f'(x)=0$ or $f'(x)$ does not exist.

Critical numbers matter because they are where a function may change from increasing to decreasing or vice versa. However, not every critical number causes a change in direction. For example, if $f'(x)$ changes from positive to positive, the function stays increasing.

Consider $f(x)=x^3$.

Its derivative is $f'(x)=3x^2$, which is $0$ at $x=0$. But $3x^2\ge 0$ for every $x$, so the function is increasing on both sides of $0$. The graph flattens at the origin, but it does not turn around.

This is a great reminder for students: a point where $f'(x)=0$ is only a candidate for a turn. The sign of the derivative on both sides decides what really happens.

Interpreting Graphs and Tables

Sometimes AP Calculus gives a graph of $f'(x)$ instead of a formula for $f(x)$. That still works because the sign of $f'(x)$ tells you the behavior of $f$.

If the graph of $f'(x)$ is above the $x$-axis, then $f'(x)>0$ and $f$ is increasing.
If the graph of $f'(x)$ is below the $x$-axis, then $f'(x)<0$ and $f$ is decreasing.
If the graph crosses the $x$-axis, that may indicate a change in increasing/decreasing behavior.

A table can also be used. If a table of values shows that $f'(x)$ changes from positive to negative, then $f$ changes from increasing to decreasing.

This is useful in real-world modeling too. For example, if $f(t)$ represents the number of users on a website over time, then $f'(t)>0$ means the user count is growing, while $f'(t)<0$ means it is shrinking.

Common Mistakes to Avoid

One common mistake is to say that $f'(x)=0$ means the function is increasing neither here nor there. That is not enough information. The correct method is to check the sign of $f'(x)$ around the point.

Another mistake is to include critical numbers inside the intervals of increase or decrease. Intervals should show where the function has consistent behavior, so critical numbers are usually excluded.

A third mistake is to confuse local behavior with global behavior. A function may be increasing on one interval and decreasing on another. Always report intervals carefully.

When solving AP problems, it helps to organize work clearly:

find $f'(x)$,
locate critical numbers,
test signs,
and write the final answer using interval notation.

Conclusion

Determining intervals where a function is increasing or decreasing is one of the clearest ways to use differentiation to understand a graph’s behavior. students, the key idea is simple but powerful: the sign of $f'(x)$ tells you whether the function rises or falls.

This skill fits into analytical applications of differentiation because it helps with graph analysis, local extrema, optimization, and interpreting function behavior from formulas, graphs, or tables. It also builds the foundation for more advanced ideas like concavity and the use of second derivatives.

If you can find critical numbers, test the sign of $f'(x)$, and explain what the result means, you are using the exact kind of reasoning AP Calculus BC expects ✅.

Study Notes

A function is increasing when $f(x)$ gets larger as $x$ increases.
A function is decreasing when $f(x)$ gets smaller as $x$ increases.
The derivative tells the slope of the graph: $f'(x)>0$ means increasing and $f'(x)<0$ means decreasing.
Find critical numbers by solving $f'(x)=0$ or identifying where $f'(x)$ does not exist, if $f(x)$ is defined there.
Use critical numbers to divide the number line into intervals.
Test the sign of $f'(x)$ on each interval to determine where the function increases or decreases.
A point where $f'(x)=0$ does not automatically mean the function changes direction.
Use open intervals when stating intervals of increase or decrease.
Graphs or tables of $f'(x)$ can also show where $f$ is increasing or decreasing.
This topic connects directly to graph analysis, local extrema, and other analytical applications of differentiation.