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Use derivatives to sketch the graph of a function

Let

    \[ f(x) = x^3 - 4x. \]

  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,

        \[ f(x) = x^3 - 4x \quad \implies \quad f'(x) = 3x^2 - 4. \]

    Thus,

        \[ f'(x) = 0 \quad \implies \quad 3x^2 - 4 = 0 \quad \implies \quad x = \pm \frac{2}{\sqrt{3}}. \]

  2. Since f'(x) > 0 when |x| > \frac{2}{\sqrt{3}} and f'(x) < 0 when |x| < \frac{2}{\sqrt{3}} we have f is increasing if |x| > \frac{2}{\sqrt{3}} and f is decreasing if |x| < \frac{2}{\sqrt{3}}.
  3. Taking the second derivative,

        \[ f'(x) = 3x^2 - 4 \quad \implies \quad f''(x) = 6x. \]

    Thus, f' is increasing for x > 0; and decreasing for x < 0.

  4. We sketch the curve,

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