Using the First Derivative Test to Determine Relative (Local) Extrema

Introduction: Why this test matters

students, when you study a graph, one of the most useful questions is: where does the function go up, where does it go down, and where does it reach a local high or low? 🌟 The First Derivative Test helps you answer that question using the sign of $f'(x)$.

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

explain what a relative maximum and relative minimum are,
use the sign of $f'(x)$ to identify where a function increases or decreases,
apply the First Derivative Test to find local extrema,
connect this idea to graph behavior and other AP Calculus AB topics.

This lesson fits into Analytical Applications of Differentiation because derivatives do more than give slopes. They also reveal the shape and behavior of a function 📈. The First Derivative Test is one of the main tools for turning derivative information into graph information.

Key ideas: extrema, critical points, and slope signs

A relative maximum is a point where a function is larger than nearby values. A relative minimum is a point where a function is smaller than nearby values. These are also called local maximum and local minimum.

A point where a function may have a local extremum is often a critical point. A critical point occurs where either $f'(x)=0$ or $f'(x)$ does not exist, as long as the point is in the domain of $f$.

The First Derivative Test focuses on what happens to the sign of $f'(x)$ around a critical point:

If $f'(x)$ changes from positive to negative, then $f$ changes from increasing to decreasing, so the function has a relative maximum.
If $f'(x)$ changes from negative to positive, then $f$ changes from decreasing to increasing, so the function has a relative minimum.
If $f'(x)$ does not change sign, then there is no relative extremum at that point.

This idea works because the sign of the derivative tells the direction of the graph:

$f'(x)>0$ means the function is increasing,
$f'(x)<0$ means the function is decreasing.

Think of walking uphill and downhill on a trail 🚶. If your slope changes from uphill to downhill, you have reached a peak. If it changes from downhill to uphill, you have reached a valley.

How to use the First Derivative Test step by step

When AP Calculus asks for relative extrema, use a careful process.

Step 1: Find the derivative

Start with the function $f(x)$ and compute $f'(x)$.

Step 2: Find critical points

Solve $f'(x)=0$ and identify any points where $f'(x)$ does not exist, if those points are in the domain.

Step 3: Make a sign chart

Choose test values in the intervals created by the critical points. Determine the sign of $f'(x)$ on each interval.

Step 4: Use the sign changes

Look at the sign change around each critical point.

positive to negative

ightarrow relative maximum

negative to positive

ightarrow relative minimum

no sign change

ightarrow neither

Step 5: State the answer clearly

Give the $x$-value of the local extremum and, if asked, the function value $f(x)$.

It is important to remember that a critical point alone does not guarantee a local maximum or minimum. The sign change is what matters.

Example 1: A polynomial with two local extrema

Consider the function

$$f(x)=x^3-3x^2-9x+1.$$

First, compute the derivative:

$$f'(x)=3x^2-6x-9.$$

Factor it:

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

Set the derivative equal to $0$:

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

so the critical points are $x=-1$ and $x=3$.

Now test the intervals $(-\infty,-1)$, $(-1,3)$, and $(3,\infty)$.

For $x=-2$, $f'(-2)=3(-5)(-1)>0$.
For $x=0$, $f'(0)=3(-3)(1)<0$.
For $x=4$, $f'(4)=3(1)(5)>0$.

So the sign pattern is $+$, then $-$, then $+$. That means:

at $x=-1$, $f'(x)$ changes from positive to negative, so there is a relative maximum,
at $x=3$, $f'(x)$ changes from negative to positive, so there is a relative minimum.

If you want the coordinates, evaluate the function:

$$f(-1)=(-1)^3-3(-1)^2-9(-1)+1=6,$$

$$f(3)=3^3-3(3^2)-9(3)+1=-26.$$

So the function has a relative maximum at $(-1,6)$ and a relative minimum at $(3,-26)$.

This is a classic AP-style problem because it combines algebra, derivatives, and interpretation of sign changes ✅.

Example 2: A derivative with repeated sign behavior

Now consider

$$g(x)=x^4.$$

Its derivative is

$$g'(x)=4x^3.$$

The critical point is $x=0$ because $g'(0)=0$.

Check the sign of $g'(x)$:

If $x<0$, then $4x^3<0$.
If $x>0$, then $4x^3>0$.

So $g'(x)$ changes from negative to positive at $x=0$. Therefore, $g$ has a relative minimum at $x=0$.

This matches the graph of $x^4$, which has a low point at the origin. Notice that the function does not have a relative maximum anywhere, because the graph keeps rising on both ends.

Example 3: A point where the derivative is zero but there is no extremum

The First Derivative Test also helps you avoid a common mistake. Consider

$$h(x)=x^3.$$

Then

$$h'(x)=3x^2.$$

The critical point is $x=0$.

However, $h'(x)=3x^2$ is never negative. It is positive for all x

eq 0$ and equals $0$ only at $x=0. So the derivative does not change sign.

That means $h$ is increasing on both sides of $x=0$, and there is no relative maximum or minimum there.

This example shows why $f'(c)=0$ alone is not enough to prove a local extremum. The sign change is the real test.

How this fits with graph analysis and other AP topics

The First Derivative Test is part of a larger toolkit for studying functions. In AP Calculus AB, you often use several ideas together:

Increasing/decreasing behavior comes from the sign of $f'(x)$.
Relative extrema come from sign changes in $f'(x)$.
Graph analysis uses derivative information to sketch and understand the shape of a curve.
The Mean Value Theorem explains why derivatives reflect average and instantaneous change.
The Extreme Value Theorem says that on a closed interval, continuous functions attain absolute maxima and minima, though the First Derivative Test is about local extrema.

Local extrema are different from absolute extrema. A function can have a local minimum without being the lowest value overall. For example, a small valley on a mountain road is a local minimum, even though the road may climb much higher elsewhere.

When solving optimization problems, you often use the First Derivative Test after finding critical points to decide which candidates give the largest or smallest values.

Common AP mistakes to avoid

Here are mistakes students often make:

forgetting to test intervals on both sides of a critical point,
assuming $f'(c)=0$ always means a max or min,
mixing up absolute and relative extrema,
ignoring points where $f'(x)$ does not exist,
forgetting to state the conclusion in words.

A strong AP answer should clearly mention the sign change of $f'(x)$ and the resulting extremum.

Conclusion

students, the First Derivative Test is one of the most important ways to connect algebraic derivative information with the shape of a graph. By finding critical points and checking whether $f'(x)$ changes from positive to negative or from negative to positive, you can determine relative maxima and minima with confidence.

This skill is central to Analytical Applications of Differentiation because it helps you analyze behavior, sketch graphs, and solve real-world problems. Whenever you see a function on an AP Calculus AB problem, remember: the derivative tells the story of how the function moves 📊.

Study Notes

A relative maximum occurs where a function changes from increasing to decreasing.
A relative minimum occurs where a function changes from decreasing to increasing.
A critical point happens where $f'(x)=0$ or where $f'(x)$ does not exist, if the point is in the domain.
The First Derivative Test uses the sign of $f'(x)$ on intervals around a critical point.
If $f'(x)$ changes from $+$ to $-$, there is a relative maximum.
If $f'(x)$ changes from $-$ to $+$, there is a relative minimum.
If $f'(x)$ does not change sign, there is no relative extremum.
A point with $f'(x)=0$ is not automatically a max or min.
This test is a key tool for graph analysis, optimization, and understanding function behavior in AP Calculus AB.