A function is twice differentiable on the interval and satisfies for every in that…
MATHEMATICS · Single correct · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A function f is twice differentiable on the interval (1,4) and satisfies f′′(x)<0 for every x in that interval. The tangent line to f at x=2.5 is used to approximate f(3). Which of the following best describes that approximation?
(a) It is less than f(3) because the graph is concave down on (1,4).
(b) It is greater than f(3) because the graph is concave down on (1,4).
(c) It is less than f(3) because x=3 is to the right of the point of tangency and the slope is decreasing.
(d) The concavity on (1,4) is insufficient to determine whether the approximation is an overestimate or an underestimate.