MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
The graph of y=f′(x), the first derivative of a twice-differentiable function f, consists of two straight-line segments and one semicircular arc. The graph intersects the x-axis at the points x=−2 and x=1. From x=−4 to x=−2, f′(x) is the line segment joining (−4,3) to (−2,0). From x=−2 to x=1, f′(x) coincides with the upper half of the circle centered at (−21,0) with radius 23. From x=1 to x=3, f′(x) is the line segment joining (1,0) to (3,2). Assume f is defined and twice differentiable for all x with f(−4)=2.
(a) For each interval determined by the x-intercepts of f′(x), determine whether f′(x) is positive or negative. Justify your conclusions with reference to the given graph description.
(b) Using your results from part (a), state the open intervals on (−4,3) where f is increasing and the open intervals where f is decreasing.
(c) Identify each relative extremum of f on the interval (−4,3), specifying the x-coordinate and whether the extremum is a maximum or a minimum. Justify your answers.
(d) On the interval [−4,3], determine the absolute maximum value of f. Clearly indicate how you reach your conclusion, including consideration of endpoints.