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Applications of Derivatives · Curve Analysis

Inflection Points

f′′(c)=0 (or DNE) and concavity changes at c⇒(c,f(c)) is an inflection pointf''(c) = 0 \text{ (or DNE) and concavity changes at } c \Rightarrow (c, f(c)) \text{ is an inflection point}

An inflection point is where the concavity changes. Find candidates where f''(x) = 0 or is undefined, then verify concavity changes.

Worked examples

Find the inflection points of f(x) = x³ - 3x² + 2.
  1. f'(x) = 3x² - 6x, f''(x) = 6x - 6
  2. f''(x) = 0 → x = 1
  3. f''(0) = -6 < 0 (concave down), f''(2) = 6 > 0 (concave up)
  4. Concavity changes at x = 1 ✓. f(1) = 0.

Answer: Inflection point at (1, 0).

Related formulas

Concavity

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