AP® Calculus BC · Unit 5 · Topic 5.7
Using the Second Derivative Test to Determine Extrema
At a critical point where f'(c)=0, check f''(c): positive means the graph curves upward into a valley (local min), negative means it curves downward into a hill (local max).
A critical point is just a flat spot: f'(c) = 0. But a flat spot can be a valley floor, a hilltop, or a flat patch that's neither. The second derivative tells you which, because f'' measures concavity — the way the curve is bending around that flat spot. If the graph is bending upward like a cup (concave up), the flat spot sits at the bottom of that cup, so it's a local minimum. If the graph is bending downward like a frown (concave down), the flat spot sits at the top, a local maximum.
Mechanically: find f'(x), solve f'(c) = 0 for critical points, then plug each c into f''(x). If f''(c) > 0, the slope is increasing through zero — negative just before c, positive just after — which is exactly the pattern for a minimum. If f''(c) < 0, the slope is decreasing through zero, the pattern for a maximum. If f''(c) = 0, the test gives no information at all; the curve might still have a max, a min, or neither (think of x4 versus x3 at x = 0). That's not a failure of calculus, it's just a question the second derivative can't answer, so you drop back to the First Derivative Test — checking the sign of f' on either side of c — or go to a higher derivative.
This test is the fast finish to the work of Unit 5: once you've found critical points from f', the second derivative lets you classify them without building a full sign chart, which matters when you're optimizing a function and need to know quickly whether you've landed on a max or a min.
Key terms
- Critical point
- A point c in the domain where f'(c) = 0 or f'(c) is undefined.
- Concavity
- The direction a graph bends: concave up if f'' > 0, concave down if f'' < 0.
- Second Derivative Test
- A rule that classifies a critical point as a local max or min based on the sign of f'' there.
- Inconclusive case
- When f''(c) = 0, the test gives no verdict and another method must be used.
Watch it explained
- How it worksHow does a positive second derivative at a critical point force a local minimum?
- Worked exampleFind and classify the extrema of f(x) = x3 - 12x using the second derivative test.
- Common mistakeWhy doesn't the second derivative test give an answer when f''(c) comes out zero?
- Going furtherWhat happens when the second derivative test fails and I fall back on the first?
Check yourself
End of topic.