A twice-differentiable function has the following key features
MATHEMATICS · Single correct · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
A twice-differentiable function f has the following key features.
• f is decreasing on (−∞,−3) and (2,∞) and increasing on (−3,2).
• f has a relative maximum at x=−3 and a relative minimum at x=2.
Which of the following descriptions is most consistent with the sign of f′?
(a) f′(x)>0 on (−∞,−3) and (2,∞), f′(x)<0 on (−3,2), and f′(x)=0 at x=−3 and x=2.
(b) f′(x)<0 on (−∞,−3), f′(x)=0 at x=−3, f′(x)>0 on (−3,2), and f′(x)>0 on (2,∞) with no additional zeros.
(c) f′(x)<0 on (−∞,−3), f′(x)=0 at x=−3, f′(x)>0 on (−3,2), f′(x)=0 at x=2, and f′(x)<0 on (2,∞).
(d) f′(x)<0 on (−∞,−3), f′(x)=0 at x=−3, f′(x)>0 on (−3,2), f′(x)=0 at x=2, and f′(x)>0 on (2,∞).