A function is defined and continuous on the interval
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A function f is defined and continuous on the interval −4≤x≤4. Its graph is described as follows.
• From x=−4 to x=−1, the graph is a straight line segment decreasing from the point (−4,3) to the point (−1,−1).
• From x=−1 to x=2, the graph is smooth and increasing, concave up on −1<x<0 and concave down on 0<x<2, passing through the point (1,0) and reaching a maximum of (2,2).
• From x=2 to x=4, the graph is a smooth, strictly decreasing, concave down curve that ends at the point (4,0).
Throughout the interval (−4,4) the curve has no cusps or corners except possibly at x=−1 and x=2, where the graph changes description.
Using only the verbal description above, analyze the behavior of the derivative f′(x).
(a) Determine all open intervals on (−4,4) where f′(x)>0, f′(x)<0, and f′(x)=0. Express each answer as an interval or set of points, and briefly justify your classifications using the geometric behavior of the graph.
(b) Identify every x–value in (−4,4) at which f′(x)=0. Explain why the derivative takes the value 0 at those points.
(c) Estimate which single point in (−4,4) is most likely to have the greatest (largest positive) value of f′(x). Support your choice with a comparison of slopes drawn from the description.
(d) Decide whether f′ exists at x=−1 and at x=2. For each point, provide a clear argument based on the definition of differentiability in terms of tangent‐line slope limits.