CalcRef

Applications of Derivatives · Curve Analysis

Second Derivative Test

f′(c)=0:f′′(c)>0⇒local min;f′′(c)<0⇒local maxf'(c) = 0: \quad f''(c) > 0 \Rightarrow \text{local min}; \quad f''(c) < 0 \Rightarrow \text{local max}

At a critical point where f'(c) = 0: if f''(c) > 0, c is a local minimum (concave up). If f''(c) < 0, c is a local maximum (concave down). If f''(c) = 0, the test is inconclusive.

Worked examples

Classify the critical points of f(x) = x⁴ - 4x².
  1. f'(x) = 4x³ - 8x = 4x(x²-2). Critical points: x = 0, x = ±√2
  2. f''(x) = 12x² - 8
  3. f''(0) = -8 < 0 → local max at x = 0
  4. f''(√2) = 24-8 = 16 > 0 → local min at x = √2
  5. f''(-√2) = 16 > 0 → local min at x = -√2

Answer: Local max at x = 0, local min at x = ±√2.

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