Take the derivative of `y=x^3+x` .
For horizontal tangent lines, the slope is zero. Set the derivative function to zero because we want to know where x is when the slope is zero.
`-1 = 3x^2`
`x^2 = -1/3`
The parabola `x^2` will never hit negative one third. We can see this on the graph below. This means we will not have a solution for x.
Therefore, there are NO POINTS where a horizontal tangent line exists.