Second Derivative Test

Business Calculus · Axiom Academy

A faster way to classify critical points The second derivative tells us about the shape of the curve: Concave Up Opens upward like a cup Concave Down Opens downward like a frown Possible Inflection Concavity may change Suppose f'(c) = 0 (c is a critical point where the derivative exists): Curve is concave down at c (like a hilltop) Curve is concave up at c (like a valley) Test fails; use First Derivative Test Negative second derivative = Negative shape = Maximum (think: sad face ⌢ at the top) Positive second derivative = Positive shape = Minimum (think: happy face ⌣ at the bottom) Example: Classify Critical Points Step 1: Find f'(x) and critical points Step 3: Evaluate f'' at each critical point Local Maximum at x = 1: f(1) = 1 - 6 + 9 + 1 = 5 Local Minimum at x = 3: f(3) = 27 - 54 + 27 + 1 = 1 Business Application: Profit Maximization Step 1: Find P'(x) and critical points Step 3: Apply Second Derivative Test Result: Maximum at x = 25 (2,500 units) Since P''(25) < 0, the function is concave down at x = 25, confirming a maximum. Maximum Profit: 450 (hundreds) = 45,000 First vs. Second Derivative Test Use Second Derivative Test when f''(c) is easy to compute and you just need to classify critical points quickly. Use First Derivative Test when the Second Derivative Test is inconclusive (f''(c) = 0), when dealing with corners or cusps, or when you need to understand the behavior around the critical point.

This is the written version of the interactive lesson above. See the full Business Calculus course.