Second Derivative Test
Concavity determines local minima and maxima.
Formula
Variables
Example
Positive = concave up = minimum
How it works
If the second derivative of a function is positive at a critical point, the graph is curving upward there, so the tangent slope is increasing as you move through the point. That upward curvature means the point sits at the bottom of a local “valley,” which is why it is a local minimum. This test comes from using curvature to refine the first-derivative test: once the slope is zero, the sign of the second derivative tells you whether the graph bends like a bowl or a cap.
When to use it
Use this when you already found a critical point and want a quick way to decide whether it is a local minimum or maximum. It is common in optimization problems in calculus, physics, economics, and any setting where you model a quantity with a differentiable function.
Worked example, step by step
Find the local minimum of f(x) = x^2 - 6x + 11. 1. Compute the first derivative: f'(x) = 2x - 6. 2. Set f'(x) = 0 to find critical points: 2x - 6 = 0 => x = 3. 3. Compute the second derivative: f''(x) = 2. 4. Evaluate at the critical point: f''(3) = 2 > 0, so the graph is concave up there. 5. Therefore, x = 3 gives a local minimum. 6. Find the minimum value: f(3) = 3^2 - 6(3) + 11 = 9 - 18 + 11 = 2. Final answer: the function has a local minimum at x = 3, and the minimum value is 2.
Common mistakes to avoid
- Using the second derivative test without first checking that the point is a critical point
- Forgetting that f''(x) > 0 indicates a local minimum, while f''(x) < 0 indicates a local maximum
- Assuming the test works when f''(x) = 0, even though the result is inconclusive in that case
Did You Know?
The second derivative measures acceleration and reveals whether a curve is concave up or down.
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