The Second Derivative Test for Local Extrema

Definition

The Second Derivative Test is a method for classifying critical points of a function. It uses the “curvature” (concavity) of the graph at a point where the slope is zero to determine if that point is a local maximum or a local minimum.

Why It Matters

The second derivative test is the ‘quick check’ for peaks and valleys; it provides the immediate confirmation needed to know if a critical point represents the highest efficiency or the lowest cost in a one-dimensional system.

Core Concepts

• Prerequisite: The test only applies at points where $f'(c) = 0$ (horizontal tangents).

• How to read: “The f-prime of c equals zero.”

  • Meaning: Critical point with horizontal tangent—candidate for max or min.

• Classification:

  • If $f''(c) < 0$, the graph is concave down, making the point a local maximum.
  • If $f''(c) > 0$, the graph is concave up, making the point a local minimum.

• How to read: “The f-double-prime of c is less than zero means local max; is greater than zero means local min.”

  • Meaning / when to use: Quick classification at critical points. Concave down = peak; concave up = valley.

• Inconclusive Result: If $f''(c) = 0$, the test fails. The point could be a maximum, a minimum, or an inflection point (e.g., $f(x) = x^4$ vs. $x^3$ vs. $-x^4$ at $x = 0$), requiring the First Derivative Test for clarification.

• How to read: “The f-double-prime of c equals zero—inconclusive.”

  • Meaning: Zero curvature at the critical point; fall back to first derivative test or higher-order analysis.

Connected Concepts

• The First Derivative Test for Local Extrema
• Second Derivative Test for Multivariable Local Extrema
• First Derivative Theorem for Local Extreme Values