Question

How do you find the points where the graph of the function `y= x +sinx` has horizontal tangents?

Answer 1

The tangent line is horizontal if and only if the slope of the tangetn line is `0`. And the slope of the tangent line is given by the derivative.

Explanation:

So, find the derivative, set it equal to `0`, and solve the resulting equation.

`y=x+sinx`

`y'=1+cosx`

`1+cosx = 0`
if and only if

`x=pi+2pik` for integer `k`.

(In words, these are the odd multiples of `pi`. They can be written `x=(2k+1)pi` for integer `k`.)

Because the sine of every such `x` is `0`, the `y` value at these point is

`y = (pi+2pik) + sin(pi+2pik) = pi+2pik` for integer `k`.

The points on the graph are: `(pi+2pik,pi+2pik)`

To help form an image of the solution above, here is the graph of the function.
(You can zoom in/out and drag the graph around to explore it a bit.)

graph{x+sinx [-25.52, 25.82, -15.62, 10.04]}