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Use derivatives to sketch the graph of a function

Let

    \[ f(x) = x^3 - 6x^2 + 9x + 5. \]

  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,

        \[ f'(x) = 3x^2 - 12x + 9. \]

    Thus,

        \[ f'(x) = 0 \quad \implies \quad 3x^2 - 12x + 9 = 0 \quad \implies \quad x = \{ 1, 3 \}. \]

  2. f is increasing for x < 1 and x > 3, and decreasing for 1 < x < 3.
  3. Taking the second derivative,

        \[ f''(x) = 6x - 12. \]

    Thus, f' is increasing if x > 2, and decreasing if x < 2.

  4. We sketch the curve,

    Rendered by QuickLaTeX.com

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