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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.

  • 2 worked examples
  • 15 in Applications of Derivatives

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).

1 more worked examplePremium

Find the inflection point of f(x) = x eˣ.

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Related formulas

Concavity

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Common questions

What is the Inflection Points?

An inflection point is where the concavity changes. Find candidates where f''(x) = 0 or is undefined, then verify concavity changes. It is one of the applications of derivatives formulas in the CalcRef reference.

How do you use the Inflection Points?

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

Practice

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