First Derivative Test

The critical points of a function are the only places where it can have a local maximum or minimum, but not every critical point is one. The first derivative test sorts them out. It watches the sign of the derivative as you pass through each critical point: if the function switches from rising to falling, you’ve found a peak; from falling to rising, a valley; and if it keeps going the same way, neither. It’s the standard way to find local max and min values, and it also gives you the intervals where the function increases and decreases.

Increasing, Decreasing and the Sign of f′

The derivative is the slope of the tangent line, so its sign tells you which way the graph is heading:

• If on an interval, is increasing there.
• If on an interval, is decreasing there.

This isn’t just a picture. It follows from the mean value theorem: for , for some between them, so whenever is positive. You may have found increasing and decreasing intervals from a graph before; the derivative lets you find them from the equation.

The First Derivative Test

First derivative test. Let be a critical point of a function that is continuous at .

• If changes from positive to negative at , then has a local maximum at .
• If changes from negative to positive at , then has a local minimum at .
• If has the same sign on both sides of , then has no local extremum at .

The reason is simple. If rises up to and falls after it, then is higher than every nearby value. That’s what a local maximum means.

How to Use the First Derivative Test

1. Find and factor it as far as you can.
2. Find the critical points: where or is undefined, with in the domain of .
3. Make a sign chart. Mark the critical points (and any points outside the domain) on a number line. They split it into intervals.
4. Test one point in each interval to find the sign of there.
5. Read the chart. + to − is a local max, − to + is a local min, no change is neither. Evaluate at each extremum to get the value.

The test points can be any numbers inside each interval. Pick easy ones.

Example 1: A Cubic

Find the local maximum and minimum of , and the intervals where is increasing and decreasing.

Step 1 — derivative.

Step 2 — critical points. at and .

Steps 3 and 4 — sign chart. Test , and :

Step 5 — read the chart.

• At , changes from + to −: local maximum .
• At , changes from − to +: local minimum .

is increasing on and , and decreasing on .

Example 2: A Critical Point That Isn’t an Extremum

Classify the critical points of .

The critical points are and . The factor is never negative, so the sign of comes from :

• At the sign stays −, so there is no extremum. The graph flattens out for an instant and keeps falling.
• At the sign changes from − to +, so there’s a local minimum .

A squared factor like in is a warning sign: the derivative touches zero there without changing sign.

Example 3: A Critical Point Where f′ Is Undefined

Find the local extrema of .

when , so . is undefined at , which is in the domain of . So the critical points are 0 and 1. Test , and :

• At , − to +: local minimum . The graph has a sharp cusp there, not a flat bottom.
• At , + to −: local maximum .

This is where the first derivative test beats the second derivative test: doesn’t exist, but the sign chart still works.

Example 4: A Function With a Gap in Its Domain

Find the local extrema of .

The critical points are . isn’t in the domain, so it isn’t a critical point, but it still goes on the sign chart because can change sign there.

• At : local maximum .
• At : local minimum .

The local maximum is smaller than the local minimum. That’s fine: “local” only compares a point with its neighbors, and the two branches of this graph are separated by the asymptote at .

Example 5: A Trig Function

Find the local extrema of on .

when , so or . Test , and :

• At : local maximum .
• At : local minimum .

Example 6: Reading the Graph of f′

AP calculus questions often show the graph of , not . Then the sign chart is already drawn for you: is positive where its graph is above the x-axis and negative where it’s below.

In the top graph, is zero at , and .

• At , the graph of crosses from above to below: + to −, so has a local maximum.
• At , it touches the axis and stays below: − to −, so neither.
• At , it crosses from below to above: − to +, so has a local minimum.

Don’t confuse the two graphs. The peaks and valleys of the graph of are not the extremes of ; they’re where changes concavity.

Can Endpoints Be Local Extrema?

In most calculus textbooks, a local maximum at means for all in some open interval around . An endpoint has no open interval around it inside the domain, so endpoints are never local extrema under that definition. Instead, you check them as candidates for the absolute maximum and minimum, using the closed interval method from the extreme value theorem lesson.

Some courses define relative extrema so that endpoints can count. If your teacher or textbook does, a sign chart still tells you: if is decreasing just after a left endpoint, that endpoint is a relative maximum.

First Derivative Test vs Second Derivative Test

The second derivative test, covered next, is faster when is easy to compute. When it gives no answer, fall back on the first derivative test.

Common Mistakes

• Using the sign of instead of . The chart is for the derivative. Whether is positive or negative doesn’t matter here.
• Forgetting critical points where is undefined, like in Example 3.
• Leaving domain gaps off the sign chart. In Example 4, could change sign at even though it isn’t a critical point.
• Calling every critical point an extremum. In Example 2, is neither.
• Giving only the x-value. The local maximum value is ; the maximum occurs at .

Practice Problems

Try each one before opening the answer. The practice set has 12 more with instant checking.

Problem 1. Find the local extremum of .

Problem 2. Find the local maximum and minimum of .

Problem 3. Does have any local extrema?

Problem 4. Find the local extremum of .

Problem 5. Classify the critical points of .

Problem 6. Suppose . Where does have local extrema?

The first derivative test reads the sign of . Next, the second derivative test classifies critical points in one step using the sign of .