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Use derivatives to sketch the graph of a function

Let

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

  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,

        \[ f'(x) = \frac{-2x + 4}{(x-1)^2 (x-3)^2}. \]

    Thus,

        \[ f'(x) = 0 \quad \implies \quad \frac{-2x+4}{(x-1)^2(x-3)^2} = 0 \quad \implies \quad x = 2. \]

  2. f is increasing if x < 1 or 1 < x < 2 and f is decreasing if 2 < x < 3 or x > 3. (We have to take some care here to leave out the points x = 1 and x = 3 since the function is not defined at these points.)
  3. Taking the second derivative,

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

    Thus, f' is increasing for x < 1 and x > 3 and f' is decreasing for 1 < x < 3.

  4. We sketch the curve,

    Rendered by QuickLaTeX.com

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