A function is twice differentiable for all real numbers except possibly at
MATHEMATICS · Single correct · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A function h is twice differentiable for all real numbers except possibly at x=2, where h′′ is undefined. The second derivative satisfies
h′′(x)<0 for x<−1,
h′′(x)>0 for −1<x<2,
h′′(x)<0 for x>2.
The first derivative satisfies h′(−3)=0,h′(0)=0, and h′(4)=0, and these are the only critical points of h.
Which of the following correctly describes the relative extrema of h and the x-values of all points of inflection of h?
(a) Relative minima at x=−3 and x=4, a relative maximum at x=0, and a point of inflection at x=−1 only.
(b) Relative maxima at x=−3 and x=4, a relative minimum at x=0, and no points of inflection because h′′ changes sign twice.
(c) Relative maxima at x=−3 and x=4, a relative minimum at x=0, and points of inflection at x=−1 and x=2.
(d) Relative maxima at x=−3 and x=4, a relative minimum at x=0, and a point of inflection at x=2 only.