Mathematics

Mathematics

Problem

Study the extrema of the trigonometric function h(x)=2sin⁡x−sin⁡2xh(x)=2\sin x-\sin 2x on the interval 0≤x≤π0\le x\le \pi. a. (i) Find h′(x)h'(x) and solve h′(x)=0h'(x)=0 to obtain all interior critical points. b. (ii) Determine h′′(x)h''(x) and apply it to classify each critical point obtained in part (i). Indicate any point at which h′′(x)=0h''(x)=0. c. (iii) Wherever the result in part (ii) is inconclusive, use the change of sign of h′(x)h'(x) to complete the classification. d. (iv) List all local maxima and minima with their coordinates and identify the greatest and least values taken by hh on [0,π][0,\pi], including the endpoints.
Intuitive explanation

To find extrema, you need to locate where the derivative equals zero—that's where the function stops increasing or decreasing. Start by differentiating h(x)h(x) using the chain rule, then solve h′(x)=0h'(x) = 0 to find critical points. Once you have those points, evaluate hh at each critical point and at the interval endpoints to compare and identify which are maxima and which are minima.

Intuitive explanation

To find extrema, you need to locate where the derivative equals zero—that's where the function stops increasing or decreasing. Start by differentiating h(x)h(x) using the chain rule, then solve h′(x)=0h'(x) = 0 to find critical points. Once you have those points, evaluate hh at each critical point and at the interval endpoints to compare and identify which are maxima and which are minima.

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