AP® Calculus AB · Unit 5 · Topic 5.7
Using the Second Derivative Test to Determine Extrema
At a critical point where f'=0, check the sign of f'': positive means the curve bends up into a valley (relative min), negative means it bends down into a peak (relative max).
A critical point is flat — f' = 0 there — but flat could mean a valley floor, a peak, or a pause on the way through. You don't need to check slopes on either side if you know how the curve bends. If f is concave up at that point, it's curving like a cup, so the flat spot sits at the bottom: a relative minimum. If f is concave down, it curves like a dome, so the flat spot is a relative maximum. Concavity alone tells you the shape without surveying neighbors.
Mechanically: find a critical point c where f'(c) = 0. Then evaluate f''(c). If f''(c) > 0, f is concave up at c, so c gives a relative minimum. If f''(c) < 0, f is concave down, so c gives a relative maximum. If f''(c) = 0, the test tells you nothing — the point could be a min, a max, or neither, so you fall back to the First Derivative Test and check sign changes of f' around c.
This test is a shortcut version of the First Derivative Test, trading a sign chart for one number. They agree whenever both apply, but the Second Derivative Test fails at flat inflection points and at any point where f'' itself doesn't exist, while the First Derivative Test always works as long as you can read signs of f'. Together they finish the job started by locating critical points: first find where f' = 0, then classify what's there, then use that classification to sketch the curve's peaks and valleys for optimization or curve-sketching problems later in the unit.
Key terms
- Critical point
- A point in the domain where f'(x) = 0 or f'(x) does not exist.
- Concavity
- The direction a curve bends: up (like a cup) when f'' > 0, down (like a dome) when f'' < 0.
- Second Derivative Test
- A rule classifying a critical point as a max or min using the sign of f'' there, inconclusive if f'' = 0.
- Relative minimum
- A point where f is smaller than at all nearby points, often the floor of a concave-up curve.
- Relative maximum
- A point where f is larger than at all nearby points, often the peak of a concave-down curve.
Watch it explained
- How it worksHow does concavity at a critical point classify it without any sign chart?
- Worked exampleFind and classify the critical points of f(x) = x3 - 12x with the second derivative test.
- Common mistakeWhy isn't the second derivative test able to decide anything when it returns zero?
- Ties togetherHow does the second derivative test compare with the first derivative test on one function?
Check yourself
End of topic.