Let on the open interval

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Let f(x)=tan⁡x−xf(x)=\tan x-x on the open interval (0,3π2)\left(0,\dfrac{3\pi}{2}\right). Respond to each of the following prompts. (a) Determine all intervals on which ff is increasing and all intervals on which ff is decreasing. Clearly show the test you use to reach your conclusions. (b) Identify each xx–coordinate in (0,3π2)\left(0,\dfrac{3\pi}{2}\right) at which ff attains a relative (local) maximum or a relative (local) minimum, and evaluate ff at those xx–values. (c) Using your results from part (a), sketch a rough graph of ff on (0,3π2)\left(0,\dfrac{3\pi}{2}\right), labeling all critical points found in part (b). No specific scale is required, but relative positions must be accurate. (d) Explain how your analysis in parts (a) and (b) verifies that the graph you sketched in part (c) is consistent with the behavior of ff.

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