Topic 5.4 Notes – Using the First Derivative Test to Determine Relative (Local) Extrema

What the First Derivative Test Says

Remember:

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

A local extremum happens when the function switches direction.

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

A critical point is where:

• $f'(c) = 0$, or
• $f'(c)$ does not exist (but $f(c)$ is defined).

Here’s the core idea:

• $f'$ changes positive → negative → local maximum
• $f'$ changes negative → positive → local minimum
• No sign change → no local extremum

You are not just checking if the derivative is zero. You are checking whether the function changes from increasing to decreasing or vice versa.

The Process in Action

When you’re given a formula and asked for relative extrema, the work follows a clear pattern.

1. Differentiate

Find $f'(x)$.

2. Find critical points

Solve $f'(x) = 0$.
Also check where $f'(x)$ is undefined.

These are the only possible locations of local extrema.

3. Make a sign chart

Place the critical points on a number line. Then test one value in each interval to determine the sign of $f'(x)$.

A typical sign chart looks like this:

Sign chart showing local minimum at x = −1 and local maximum at x = 2

From this chart:

• At $x = -1$: negative → positive → local minimum
• At $x = 2$: positive → negative → local maximum

That sign change is your justification.

On FRQs, graders want to see wording like:
“Since $f'$ changes from positive to negative at $x = 2$, $f$ has a local maximum at $x = 2$.”

Multiplicity and Factored Derivatives

If $f'(x)$ is factored, you can predict sign changes without plugging in tons of numbers.

Example structure:

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

Two rules save time:

• Odd power factor → sign changes at that zero
• Even power factor → sign does not change

So here:

• $x = 3$ (power 2) → no sign change → not an extremum
• $x = -1$ (power 1) → sign changes → extremum

This shows up a lot in multiple choice. If you ignore multiplicity, you’ll misclassify points.

Special Situations

Derivative equals zero but no extremum

If the derivative is zero and the sign does not change, the graph flattens but keeps moving in the same direction.

That gives a horizontal tangent, not a max or min.

A classic example is $y = x^3$ at $x = 0$:

$y = x^3$ with a horizontal tangent at (0,0)

Derivative undefined

If $f'(c)$ does not exist, still test the sign on both sides.

Corners and cusps can be local extrema if the sign changes.

What This Test Does and Does Not Do

The First Derivative Test finds local extrema only.

It does not guarantee absolute extrema unless you also check endpoints on a closed interval. That’s a different process from earlier in Unit 5.

This topic is about justifying behavior using derivatives. You are connecting:

• sign of derivative
• increasing/decreasing behavior
• existence of turning points

That chain of reasoning is what the AP exam cares about.