What values of x of the graph have a horizontal tangent?

Question

For what values of x does the graph of f(x) = x + 2 sin x

 have a horizontal tangent? (Use n as your integer variable. Enter your answers as a comma-separated list.)

Answer 1

A horizontal line has a slope of zero.

For f(x) = x + 2sin(x) to find the derivative we use the power rule and the rule that for y = sin(x) then y' = cos(x).

So f '(x) = 1 + 2cos(x)

Then to find the horizontal tangent, set f '(x) equal to zero:

1 + 2cos(x) = 0

cos(x) = -1/2

x = 2π/3 or x = 4π/3 but this is true for every time you go around the unit circle so x = 2π/3 + 2πn and x = 4π/3 + 2πn