A twice–differentiable function p has derivative values summarized in the table below. Assume p′ and p′′ are continuous where defined.
x
1
2
3
4
p′(x)
0
0
0
0
p′′(x)
−5
0
7
−3
No other points satisfy p′(x)=0 in the closed interval [1,4].
(a) For each x–value in the table, indicate whether p has a local maximum, local minimum, or whether the available information is inconclusive. Give precise reasoning for every decision.
(b) Choose one x–value where your answer in part (a) was "inconclusive." Describe in detail an additional analytic or graphical investigation that would resolve the nature of the critical point at that x–value. Explain why your proposed investigation would succeed.