Study the extrema of the trigonometric function on the interval .

MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives

Problem

Study the extrema of the trigonometric function

h(x)=2\sin x-\sin 2x

on the interval

0\le x\le \pi

. (i) Find

h'(x)

and solve

h'(x)=0

to obtain all interior critical points. (ii) Determine

h''(x)

and apply it to classify each critical point obtained in part (i). Indicate any point at which

h''(x)=0

. (iii) Wherever the result in part (ii) is inconclusive, use the change of sign of

h'(x)

to complete the classification. (iv) List all local maxima and minima with their coordinates and identify the greatest and least values taken by

h

[0,\pi]

, including the endpoints.