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Calculus · Extrema

Critical points of x·e^(-x)

x e^{- x}

Step by step

  1. f(x) = x e^{- x}

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

  2. f'(x) = - x e^{- x} + e^{- x}

    Differentiate.

  3. x = 1

    Solve f′(x) = 0.

  4. f''(x) = x e^{- x} - 2 e^{- x}

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

  5. f''(1) = - \frac{1}{e} \Rightarrow (1, e^{-1}) \text{ is a local maximum}

Reveal the answer
(1, e^{-1})\ \text{local maximum}