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Activity 3.5.2.
Let
\(g(x) = \frac{1}{3}x^3 - 2x + 2\text{.}\)
(a)
Find all critical numbers of
\(g\) that lie in the interval
\(-2 \le x \le 3\text{.}\)
(b)
Use a graphing utility to construct the graph of
\(g\) on the interval
\(-2 \le x \le 3\text{.}\)
(c)
From the graph, determine the
\(x\)-values at which the absolute minimum and absolute maximum of
\(g\) occur on the interval
\([-2,3]\text{.}\)
(d)
How do your answers change if we instead consider the interval
\(-2 \le x \le 2\text{?}\)
(e)
What if we instead consider the interval
\(-2 \le x \le 1\text{?}\)
