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Use derivatives to sketch the graph of a function

Let

    \[ f(x) = \frac{1}{6}x^2  + \frac{1}{12} \cos (2x). \]

  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,

        \[ f'(x) = \frac{1}{3}x - \frac{1}{6} \sin (2x). \]

    Thus,

        \[ f'(x) = 0 \quad \implies \quad \frac{1}{3} x - \frac{1}{6} \sin (2x) = 0 \quad \implies \quad x = 0. \]

  2. f is increasing if x > 0 and decreasing if x < 0.
  3. Taking the second derivative,

        \[ f''(x) = \frac{1}{3} - \frac{1}{2} \cos (2x). \]

    Thus, f' is increasing for all x.

  4. We sketch the curve,

    Rendered by QuickLaTeX.com

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