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Use derivatives to sketch the graph of a function

Let

    \[ f(x) = x^2 - 3x + 2. \]

  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,

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

    Thus,

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

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

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

    Since this is positive for all x, this means f' is increasing for all x.

  4. We sketch the curve,

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