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

1. What the First Derivative Test Says About Local Extrema

Everything starts with a critical point.

A critical point happens where:

• $f'(x) = 0$, or
• $f'(x)$ is undefined (and $f$ exists there).

Now remember what the derivative tells you:

• $f'(x) > 0$ → function is increasing
• $f'(x) < 0$ → function is decreasing

The First Derivative Test uses this idea:

• If $f'$ changes positive → negative, the function goes increasing → decreasing → local maximum
• If $f'$ changes negative → positive, the function goes decreasing → increasing → local minimum
• If $f'$ does not change sign, there is no local extremum

Think in motion:

• Climbing then descending = hill (max)
• Descending then climbing = valley (min)

Here’s what that sign change looks like on a sign chart:

First Derivative sign chart

At point a, the sign changes from positive to negative, so there is a local maximum. At b, the sign stays negative on both sides, so there is no extremum. At c, the sign changes from negative to positive, so there is a local minimum.

Notice that what matters is the sign change, not just whether $f'(x) = 0$.

2. The Full Classification at a Critical Point

At any critical point, only four things can happen:

On tests, you are not guessing from the graph. You are justifying using derivative behavior.

A correct AP-style sentence sounds like:

Since $f'(x)$ changes from negative to positive at $x = 2$, $f$ has a relative minimum at $x = 2$.

That explanation earns the point.

3. How to Apply the First Derivative Test

Let’s walk through the process cleanly.

Step 1: Differentiate

Find $f'(x)$.

Step 2: Find Critical Points

Solve:

• $f'(x) = 0$
• Where $f'(x)$ is undefined (but $f$ is defined)

These are the only possible locations of local extrema.

Step 3: Create a Sign Chart

Place critical points on a number line.

Then:

• Either plug in test values
• Or analyze signs from factored form

For example, if

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

Critical points: $x=1$, $x=-3$

Test intervals:

• Left of −3 → both factors negative → positive
• Between −3 and 1 → one positive, one negative → negative
• Right of 1 → both positive → positive

Sign pattern: $+ \to - \to +$

So:

• $x=-3$: $+ \to -$ → local max
• $x=1$: $- \to +$ → local min

That full reasoning is what earns FRQ credit.

4. Patterns That Show Up Often on Tests

Factored derivatives

When $f'(x)$ is already factored, use sign logic instead of plugging numbers. It’s faster and cleaner.

Repeated factors (multiplicity)

If

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

The squared factor has even multiplicity, so its sign does not change.

Rule:

• Even power → no sign change
• Odd power → sign changes

This lets you classify points quickly without test values.

When a graph of $f'$ is given

Very common on AP exams. Instead of an equation, you get the graph of $f'(x)$ and must decide what $f$ is doing.

Graph of $f'(x)$ showing sign changes

Notice in the graph above:

• At $x=-2$, $f'$ crosses from positive to negative → local maximum of $f$
• At $x=1$, $f'$ touches 0 but does not change sign → not an extremum
• At $x=3$, $f'$ crosses from negative to positive → local minimum of $f$

You are analyzing sign change, not height of the graph.

5. Common Mistakes That Cost Points

• Saying “$f'(a)=0$, so there’s a max/min.” That is incomplete. You must show a sign change.
• Checking only one side of the critical point.
• Mixing up max and min. If you forget, picture walking along the graph.
• Forgetting this only gives local extrema. Absolute extrema require checking endpoints on closed intervals.

The entire idea of this topic is simple but powerful:

The behavior of the derivative determines the behavior of the function.