# `f(x) = sin(x) + cos(x)` Consider the function on the interval (0, 2pi). Find the open intervals on which the function is increasing or decreasing, apply first derivative test to identify all relative extrema.

The function f(x) = sin x + cos x.

In the open interval `(0, 2*pi)` , the intervals in which the function is decreasing and increasing has to be identified.

For a function f(x), if f'(x) >0, the function is increasing at x; and if f'(x)<0, it is decreasing at x. At values of x where f'(x) = 0, the function f(x) has relative extrema.

For f(x) = sin x + cos x

f'(x) = cos x - sin x

cos x - sin x = 0

=> sin x = cos x

=> tan x = 1

`x = tan^-1(1)`

In the interval `(0,2*pi)` , x can take on the values `{pi/4, 5*pi/4}`

Now there are 3 intervals, `(0, pi/4)` , `(pi/4, 5*pi/4` ), (`5*pi/4, 2*pi)`

In the interval `(0, pi/4)` , the value of f'(x) is positive.

In the interval `(pi/4, (5*pi)/4)` , the value of f'(x) is negative

In the interval...

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