Find the intervals in which the function f given by f(x) = sinx + cosx ,o ≤ x ≤ 2π

If you want to study the monotony of the function, we'll
have to use the derivative of the function. We'll differentiate the function with
respect to x:

f'(x) = cos x -
sinx

Now, we'll cancel
f'(x):

`f'(x) = 0 lt=gt cos x - sinx = 0 =gt 1 - sin x/cos
x = 0`

But `sinx/cosx = tan
x`

1 - tan x = 0

tan x =
1

The values of the tangent function are positive within
the 1st and the 3rd quadrants.

Therefore, we'll
have:

`x = pi/4`

`` `x = pi +
pi/4`

`` `x = (5pi)/4`

We
notice that the tangent function is increasing over the interval (0,pi/2), therefore the
function f(x) is increasing over (0,pi/2). Since the values of the tangent function are
negative over (pi/2,pi), the derivative f'(x) is decreasing over this interval and the
function f(x) is also decreasing over the interval
(pi/2;pi).

Therefore, the function is
increasing over (0;pi/2) and (pi ; (3pi)/2) and it is decreasing over (pi/2 ; pi) and
((3ipi)/2 ; 2pi).