The derivative of a function has a critical point at (that is, )
MATHEMATICS · Single correct · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
The derivative of a function f has a critical point at x=1 (that is, f′(1)=0). Selected values of the second derivative are
f′′(0)=4,f′′(1)=0,f′′(2)=−3.
Which conclusion about the behavior of f at x=1 is most accurate?
(a) f has a relative maximum at x=1 because f′′ changes from positive to negative at that point.
(b) The information given is insufficient to determine the behavior of f at x=1.
(c) f has neither a relative maximum nor a relative minimum at x=1 because f′′(1)=0 and the sign of f′′ changes across x=1.
(d) f has a relative minimum at x=1 because f′′(1)=0.