The function is defined on the closed interval by .
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
The function h is defined on the closed interval [0,2π] by h(x)=2sin2x−3x.
(a) Write an explicit formula for h′(x).
(b) Solve h′(x)=0 for x in the open interval (0,2π), giving exact values.
(c) Determine all subintervals of (0,2π) on which h is increasing and all subintervals on which h is decreasing. Provide analytic justification.
(d) Evaluate h at the critical points found in part (b) and at the endpoints of the interval. Using these values together with your results from part (c), identify the absolute maximum and absolute minimum values of h on [0,2π], stating both the x–coordinates and the corresponding function values. Explain your reasoning thoroughly.