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Use derivatives to sketch the graph of a function

by RoRi

Let

    \[ f(x) = \frac{x^2 - 4}{x^2 - 9}. \]

  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,

        \[ f'(x) = \frac{-10x}{(x^2 - 9)^2}. \]

    Thus,

        \[ f'(x) = 0 \quad \implies \quad \frac{-10x}{(x^2 -9)^2} = 0 \quad \implies \quad x = 0. \]

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

        \[ f''(x) = \frac{30(x^2+3)}{(x^2-9)^3}. \]

    Thus, f' is increasing for x < -3 or x > 3 and is decreasing if -3 < x < 3.

  4. We sketch the curve,

    Rendered by QuickLaTeX.com

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