For what values of x does the graph of f have a horizontal tangent? (Use n as your integer variable. Enter your answers as a comma-separated list.)

f (x) = x - 2 sin x

F (x) = x - 2sinx

To find for what values of x graph has tangents we need to differentiate function f (x).

f' (x) = 1-2cosx

Now, if we want to have horizontal tangents we need to make function f' (x) equal to 0 because at minimum or maximum of function f (x) tangent is horizontal.

1-2cosx = 0

cosx = 1/2

x = pi/3 + 2*n*pi and

x = - pi/3 + 2*n*pi

Explanation: cosx = 1/2 for both 60° and - 60°. If we make one circle cosx will be equal to 1/2 for 420 and 300 degrees. Writing degrees in radians (using pi) we can write formulas above where 2*pi represents one full circle and n is an integer which represents how many circles we made.