MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
A twice-differentiable function q is defined on [−5,8]. The graph of y=q′(x) consists of five semicircles of radius 1 and a constant segment, described as follows.
• Semicircle 1 is above the x-axis, centered at x=−4.
• Semicircle 2 is below the x-axis, centered at x=−2.
• Semicircle 3 is above the x-axis, centered at x=0.
• Semicircle 4 is below the x-axis, centered at x=2.
• Semicircle 5 is above the x-axis, centered at x=4.
• From x=5 to x=8, q′(x)=−1.
Each semicircle meets the x-axis exactly at its endpoints. Assume q(−5)=0.
(a) Compute the exact signed area contributed by each semicircular segment to the net change in q over its span.
(b) Using the areas from part (a), evaluate q(x) at x=−3,−1,1,3,5, and 8.
(c) List every point in [−5,8] that could yield an absolute extremum of q. Determine which of these points give the absolute maximum and the absolute minimum values of q, and state those values.
(d) Provide a complete justification for the answers in part (c), citing both the calculated function values and the behavior of q′ on intervals where q was not explicitly evaluated.