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

1 Answer

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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).