Activity 3.1.2.
Suppose that \(g(x)\) is a function continuous for every value of \(x \ne 2\) whose first derivative is \(g'(x) = \frac{(x+4)(x-1)^2}{x-2}\text{.}\) Further, assume that it is known that \(g\) has a vertical asymptote at \(x = 2\text{.}\)
(a)
Determine all critical numbers of \(g\text{.}\)
(b)
By developing a carefully labeled first derivative sign chart, decide whether \(g\) has as a local maximum, local minimum, or neither at each critical number.
(c)
Does \(g\) have a global maximum? global minimum? Justify your claims.
(d)
What is the value of \(\displaystyle \lim_{x \to \infty} g'(x)\text{?}\) What does the value of this limit tell you about the long-term behavior of \(g\text{?}\)
(e)
Sketch a possible graph of \(y = g(x)\text{.}\)
