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Use derivatives to sketch the graph of a function

Let

    \[ f(x) = x - \sin x. \]

  1. Find all points such that f'(x) = 0;
  2. Determine the intervals on which f is monotonic by examining the sign of f';
  3. Determine the intervals on which f' is monotonic by examining the sign of f'';
  4. Sketch the graph of f.

  1. We take the derivative,

        \[ f'(x) = 1 - \cos x. \]

    Thus,

        \[ f'(x) = 0 \quad \implies \quad 1 - \cos x = 0 \quad \implies \quad x = 2n \pi. \]

  2. f is increasing for all x since 1 - \cos x \geq 0 for all x.
  3. Taking the second derivative,

        \[ f''(x) = \sin x. \]

    Thus, f' is increasing for 2 n \pi < x < (2n+1) \pi and decreasing for (2n-1) \pi < x < 2n \pi.

  4. We sketch the curve,

    Rendered by QuickLaTeX.com

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