Question

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

Answer 1

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.