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Calculus · Extrema

Critical points of sin(x)

\sin{\left(x \right)}

Steg för steg

  1. f(x) = \sin{\left(x \right)}

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

  2. f'(x) = \cos{\left(x \right)}

    Differentiate.

  3. x = \frac{\pi}{2}, x = \frac{3 \pi}{2}

    Solve f′(x) = 0.

  4. f''(x) = - \sin{\left(x \right)}

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

  5. f''(\frac{\pi}{2}) = -1 \Rightarrow (\frac{\pi}{2}, 1) \text{ is a local maximum}

  6. f''(\frac{3 \pi}{2}) = 1 \Rightarrow (\frac{3 \pi}{2}, -1) \text{ is a local minimum}

Avslöja svaret
(\frac{\pi}{2}, 1)\ \text{local maximum},\; (\frac{3 \pi}{2}, -1)\ \text{local minimum}