- #1

Cosmophile

- 111

- 2

## Homework Statement

Identify the intervals of increase/decrease of ##f(x) = \sin x + \cos x##

## Homework Equations

##f(x) = \sin x + \cos x##

##f'(x) = \cos x - \sin x = \sqrt 2 \cos(x+\frac {\pi}{4})##

## The Attempt at a Solution

##f## is increasing when ##f'(x) > 0##

##\sqrt 2 \cos(x+\frac {\pi}{4}) > 0 \to \cos(x + \frac {\pi}{4}) > 0##

So, ##f## is increasing on the intervals ##(0, \frac {\pi}{4})## and ##(\frac {5 \pi}{4}, 2\pi)##, and is decreasing on the interval ##(\frac {\pi}{4}, \frac {5 \pi}{4})##.

I know that in order to generalize this, you would add ##2 \pi n## to all intervals. I simply would like to know how this would be mathematically written. Thanks!