Determine the number(s), x, between 0 and 2π where the line tangent to the function f(x)=2sin(x)+2cos(x) is horizontal.

The line tangent to (the graph of) a function is horizontal, when it's slope is equal to 0.

The slope of the tangent line at a point, is determined by the derivative of the function, at that point.

So, we find the derivative of f, and find when it is equal to 0:

f'=2cos(x)-2sin(x)

2cos(x)-2sin(x)=0

cos(x)=sin(x), 

then x is 45 degrees or 45+180=225 degrees, which are

radians, and [tex] \displaystyle{\frac{ 5\pi }{4}

 π/4, 5π/4