15.3 The Second Derivative Test

The second derivative test is a powerful tool for finding and classifying extrema of functions. It uses the sign of the second derivative at critical points to determine if they're local maxima or minima.

This method is faster than the first derivative test, only requiring evaluation at the critical point itself. However, it can be inconclusive in some cases, necessitating further analysis or use of other techniques.

The Second Derivative Test

Second derivative test conditions

• Function $f(x)$ must be twice differentiable at the critical point $c$ means $f''(c)$ exists
• Critical point $c$ satisfies $f'(c) = 0$ first derivative equals zero at $c$
• $f''(c) < 0$ indicates $c$ is a local maximum (concave down at $c$)
• $f''(c) > 0$ indicates $c$ is a local minimum (concave up at $c$)
• $f''(c) = 0$ test is inconclusive, $c$ may be a local maximum, local minimum, or neither (inflection point, $f(x) = x^3$)

Classification of critical points

• Find critical points by solving $f'(x) = 0$ (stationary points, $f(x) = x^2$, $c = 0$)
• Evaluate second derivative $f''(x)$ at each critical point
• Classify critical points based on sign of $f''(c)$:
  • $f''(c) < 0$ local maximum (peak, $f(x) = -x^2$, $c = 0$)
  • $f''(c) > 0$ local minimum (valley, $f(x) = x^2$, $c = 0$)
  • $f''(c) = 0$ inconclusive, further analysis needed (inflection point, $f(x) = x^3$, $c = 0$)

First vs second derivative tests

• First derivative test:
  • Uses sign changes of $f'(x)$ around critical point to classify
  • Requires evaluating $f'(x)$ left and right of critical point
  • Classifies critical points as local max, min, or neither (saddle point, $f(x,y) = x^2 - y^2$, $(0,0)$)
• Second derivative test:
  • Uses sign of $f''(x)$ at critical point to classify
  • Requires evaluating $f''(x)$ only at critical point
  • Classifies critical points as local max or min, may be inconclusive
• Both require differentiability at critical point
• Second derivative test faster and more direct, first derivative test handles inconclusive cases

Applications in optimization problems

1. Identify objective function $f(x)$ to maximize or minimize (profit, cost, area)

2. Find critical points by solving $f'(x) = 0$

3. Apply second derivative test to classify critical points:

   • For maximization, choose critical point with $f''(c) < 0$ (max profit, max volume)
   • For minimization, choose critical point with $f''(c) > 0$ (min cost, min surface area)

4. If inconclusive, use first derivative test or analyze function behavior around critical points

5. Interpret results in context of original problem (optimal price, dimensions)