MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
On the closed interval 0≤x≤2π, define h(x)=x+sin(2x).
(1) Derive an explicit expression for h′(x) and determine all x in [0,2π] where h′(x)=0. (Include endpoints when appropriate.)
(2) Analyze the sign of h′(x) to state precisely where h is increasing and where it is decreasing on [0,2π]. Clearly indicate any open or closed intervals used.
(3) Classify each interior critical point from part (a) as a relative minimum or relative maximum. Support your answer with reasoning tied to the behavior of h′(x).
(4) Among the relative extrema and the endpoint values of h, identify which x gives the absolute maximum and which gives the absolute minimum of h on [0,2π]. Justify your selections.
(5) List all interior relative extrema in order of their function values, from least to greatest.