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Calculus · Extrema

Critical points of x^3 - 12x

x^{3} - 12 x

Step by step

  1. f(x) = x^{3} - 12 x

    Critical points are where f′(x) = 0 or is undefined.

  2. f'(x) = 3 x^{2} - 12

    Differentiate.

  3. x = -2, x = 2

    Solve f′(x) = 0.

  4. f''(x) = 6 x

    Second-derivative test: f″ < 0 → maximum, f″ > 0 → minimum.

  5. f''(-2) = -12 \Rightarrow (-2, 16) \text{ is a local maximum}

  6. f''(2) = 12 \Rightarrow (2, -16) \text{ is a local minimum}

Reveal the answer
(-2, 16)\ \text{local maximum},\; (2, -16)\ \text{local minimum}