Example Problem Stated

Using first and second derivative test, determine the absolute extrema of the function \[f(x) = \frac{1}{5}\left(x -3\right)^2\left(x + \frac{1}{2}\right)^3\]over the interval \(\left[-1,\frac{7}{2}\right]\).

Reminder:

First Derivative Test: Suppose that \(c\) is a critical number of a continuous function \(f\).

If \(f'(x)\) changes from negative to positive at \(x = c\), then \(f(x)\) has a local minimum at \(x = c\).
If \(f'(x)\) changes from positive to negative at \(x = c\), then \(f(x)\) has a local maximum at \(x = c\).
If \(f'(x)\) does not change signs at \(x = c\), then \(f(x)\) does not have a local extrema at \(x = c\).

Second Derivative Test: Suppose that \(c\) is a critical number of a continuous function \(f\).

If \(f'(c) = 0\) and \(f''(c)>0\), then \(f(x)\) has a local minimum at \(x = c\).
If \(f'(c) = 0\) and \(f''(c)<0\), then \(f(x)\) has a local maximum at \(x = c\).
If \(f'(c) = 0\) and \(f''(c)=0\), then you must fall back to the first-derivative test to draw any conclusions.

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