Need help with a Calculus question.

Question

For what value(s) of x does the graph of the function f(x)=x+2 cos x have a horizontal tangent line? Justify your answer analytically (that is, without the use of graphing software)

Answer 1

Horizontal tangent line has slope m = 0. But m = f' = 0;

f' = 1 - 2sinx = 0; sinx = 1/2; x = π/6 + 2πn or x = 5π/6 + 2πn

At these points tangent lines horizontal