A twice-differentiable function satisfies the following properties of its second…
MATHEMATICS · Single correct · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
A twice-differentiable function f satisfies the following properties of its second derivative:
• f′′(x)=0 at x=−4,−2,1, and 5 only.
• f′′(x)>0 on (−∞,−4),(−2,1), and (5,∞).
• f′′(x)<0 on (−4,−2) and (1,5).
In addition, the first derivative satisfies f′(−3)=0 and f′(4)=0, and these are the only values of x for which f′(x)=0.
Which statement is consistent with the information above?
(a) f has relative minima at x=−3 and x=4, and points of inflection at x=−4,−2,1, and 5.
(b) f has a relative maximum at x=−3 and a relative minimum at x=4, and points of inflection at x=−4,−2,1, and 5.
(c) f has relative maxima at x=−3 and x=4, and points of inflection at x=−4,−2,1, and 5.
(d) f has relative maxima at x=−3 and x=4, and points of inflection only at x=−2 and x=1.