Suppose the first derivative of a differentiable function satisfies the following…
MATHEMATICS · Single correct · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
Suppose the first derivative of a differentiable function f satisfies the following conditions:
• f′ is decreasing on (−5,2).
• f′ is increasing on (2,7).
Which statement most accurately describes the behavior of the graph of f at x=2?
(a) The graph of f has a relative minimum at x=2 because f′ changes from decreasing to increasing, so it must cross 0.
(b) The graph of f is concave up on both sides of x=2 since f′ is monotonic throughout the interval.
(c) The graph of f changes concavity at x=2 because the monotonicity of f′ switches from decreasing to increasing there.
(d) The graph of f has neither a relative extremum nor an inflection point at x=2 because f′′(2)=0 does not guarantee any change in behavior.