The function is continuous on the open interval and differentiable everywhere on that…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
The function g is continuous on the open interval (−4,5) and differentiable everywhere on that interval except possibly at x=1. A portion of a sign chart for g′(x) is summarized below.
(Information is read left-to-right.)
• For −4<x<−2, g′(x)>0.
• At x=−2, g′(−2)=0.
• For −2<x<1, g′(x)<0.
• At x=1, g′ does not exist.
• For 1<x<3, g′(x)<0.
• At x=3, g′(3)=0.
• For 3<x<5, g′(x)>0.
Assume no other critical behavior occurs in (−4,5).
(1) Identify all critical points of g in the interval (−4,5). Give numerical x-values only.
(2) For each critical point listed in part (a), determine whether g has a local maximum, a local minimum, or neither at that point. Justify each classification with reference to the behavior of g′(x) on the intervals between critical points.