Draw the graph of the function `f(x)=max dot {(sinx ,cos2x)},x in [0,2pi]dot` Write the equivalent definition of `f(x)` and find the range of the function.

Explanation

We have `f(x) = max. {sin x, cos 2x], x in [0, 2pi]`
We first draw the graphs of y = sin x and y = cos 2x.

`rArr` `sin x = 1 - 2 sin^(2) x`
`rArr` `2 sin^(2)x + sin x - 1 = 0`
`rArr` `(2 sin x - 1)(sin x + 1) = 0`
`rArr` `sin x = 1//2` or `sin x = -1`
`rArr` `x = pi//6, 5pi//6` or `x = 3pi//2`
`f(x) = {{:(cos 2x",", 0 le x lt (pi)/(6)),(sin x",", (pi)/(6) le x lt (5pi)/(6)),(cos 2x",", (5pi)/(6) lt x le 2pi):}`
Also the range of the function is [-1, 1].