Question

f(x) = cos(x) - (cos(x))^2 for -pi is less than or equal to x is less than or equal to pi

1) x-intercepts?
2) x and y coordinates of all relative maximum points?
3) intervals on which f is increasing?

then i have to sketch f, so tips on doing so would be appreciated. thank you!

Answer 1

I assume you know how to plot a graph. Choose 10 or more x values between -pi and pi, compute f(x) for each, and graph them with f(x) on the y axis. Draw a smooth line between the points.

The x intercepts are where y = 0. That would be whenever cos x = 0 or 1. You should know where those angles are x = 0 and x = pi/2 are two of them.

Compute the derivative f'(x) and set it equal to zero. The solutions of that equation will be where the function has a relative maximum or minimum.
f'(x) = -sin x + 2 sin x cos x
= sin x (2 cos x -1)
which equals 0 at x = 0, x = pi/2 and cos^1(1/2). Only the last of those is a maximum.
f is increasing where f'(x) is positive. Drawing the graph should help you find that region.