MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A twice-differentiable function f is defined for 0≤x≤3. Table 2 gives selected numerical values of f′′ at unequally spaced x-values.
Table 2. Values of f′′(x)
x 0 0.5 1.2 2.0 3.0
f''(x) 3.2 1.8 0.5 −0.4 −1.6
Respond to each part. Support every conclusion with appropriate justification that refers to the data in Table 2.
(a) Indicate the sign of f′′ on each subinterval determined by the x-values in the table.
(b) Determine all open intervals on 0<x<3 where the graph of f is concave up and all open intervals where it is concave down.
(c) Estimate, to the nearest 0.1 unit, the x-coordinate of every point of inflection of f on the closed interval [0,3]. Explain your procedure for obtaining each estimate.