# `y = x^3 + x` Determine the point(s) (if any) at which the graph of the function has a horizontal tangent line.

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Expert Answers

kalau | Certified Educator

Take the derivative of `y=x^3+x` .

`y'= 3x^2+1`

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.

`0=3x^2+1`

`-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.