The graph of is composed of two semicircles and two line segments on the closed interval
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
The graph of y=f′(x) is composed of two semicircles and two line segments on the closed interval −3≤x≤5.
• From x=−3 to x=−1, f′(x) traces the upper half of a circle of radius 2 centered at (−1,0).
• From x=−1 to x=1, f′(x) is the line segment connecting (−1,0) to (1,0).
• From x=1 to x=3, f′(x) traces the lower half of a circle of radius 2 centered at (3,0).
• From x=3 to x=5, f′(x) is the line segment with slope −1 connecting (3,0) to (5,−2).
Assume f(0)=7.
Respond comprehensively to all parts.
(a) Compute the exact value of f(−3). Show all calculations leading to your answer.
(b) Determine the absolute maximum value of f on the closed interval −3≤x≤5. State both the x-coordinate at which it occurs and the maximum value itself, providing full justification.
(c) Identify every x-value in −3≤x≤5 at which the graph of y=f(x) has a point of inflection. Explain your reasoning.
(d) Sketch a possible graph of y=f(x) over the interval, clearly marking all extrema and inflection points found in parts (b) and (c).