Activity 3.3.3.
Suppose that \(g\) is a function whose second derivative, \(g''\text{,}\) is given by the graph in the following figure.
(a)
Identify all \(x\)-values where \(g''(x) = 0\) or \(g''(x)\) is undefined. Then, construct a second derivative sign chart for \(g\) and hence state the intervals on which \(g\) is concave up, as well as the intervals on which \(g\) is concave down.
(b)
(c)
Suppose you are given that \(g'(-1.67857351) = 0\text{.}\) Is there is a local maximum, local minimum, or neither (for the function \(g\)) at this critical number of \(g\text{,}\) or is it impossible to say? Why?
(d)
Assuming that \(g''(x)\) is a polynomial (and that all important behavior of \(g''\) is seen in the graph above), what degree polynomial do you think \(g(x)\) is? Why?
