AP® Calculus AB · Unit 5 · Topic 5.4
Using the First Derivative Test to Determine Relative Extrema
The First Derivative Test classifies a critical point by checking whether f prime changes sign around it: positive to negative means a relative maximum, negative to positive means a relative minimum, and no change means neither.
A relative peak is a place where the curve stops climbing and starts falling. That means the slope itself flips from positive to negative right at that point. A relative valley is the mirror image: the slope goes from negative to positive. So instead of staring at the curve, you can just track the sign of f' as x moves left to right through a critical point.
To run the test, find every critical point first: places where f'(x) = 0 or where f'(x) does not exist. Then pick a test value on each side of that point and check the sign of f' there. If f' goes from positive to negative, you have a relative maximum. If it goes from negative to positive, you have a relative minimum. If the sign stays the same on both sides, the point is neither — the function was already increasing (or decreasing) straight through it, it just paused or had a corner.
This test is the practical partner to the idea of critical points from earlier in the unit: critical points are the only candidates for relative extrema, but not every candidate gets the job. It also sets up the Second Derivative Test, which asks about concavity instead of sign changes, and it feeds directly into curve sketching and optimization, where you need to know not just where the extrema could be, but which ones actually are maxima or minima.
Key terms
- Critical point
- A point in the domain of f where f'(x) = 0 or f'(x) does not exist.
- Relative maximum
- A point where f changes from increasing to decreasing, so f' goes from positive to negative.
- Relative minimum
- A point where f changes from decreasing to increasing, so f' goes from negative to positive.
- First Derivative Test
- A method that classifies each critical point by checking whether f' changes sign around it.
- Sign change
- A switch in whether f' is positive or negative as x crosses a critical point, revealing the shape of the curve there.
Watch it explained
- How it worksHow does the first derivative test read a sign change as a maximum or a minimum?
- Worked exampleFind and classify the relative extrema of f(x) = x4 - 8x2 + 3 with the first derivative test.
- Common mistakeWhy isn't a critical point with no sign change an extremum at all?
- Going furtherWhat does the test say at a critical point where the derivative does not exist?
Check yourself
End of topic.