Question

How do you determine at which the graph of the function `y=x^3+x` has a horizontal tangent line?

Answer 1

Set the derivative equal to `0` and solve.

Explanation:

A line is horizontal if and only if its slope is `0`.

The derivative of a function gives (a formula for) the slope of the line tangent to the graph of the function.

Therefore, the line tangent to the grapg of a function is horizontal at every value of `x` that makes the derivative equal to `0`.

`f(x)=x^3+1`

`f'(x) = 3x^2+1`

`f'(x) = 0` at solution(s) to `3x^2+1=0`.

`3x^2+1=0` if and only if `x^2 =-1/3`, which is not possible for real number values of `x`.

There are no tangent lines that are horizontal.

Bonus answer

Because `x^2 >= 0` for all real `x`,

`f'(x) >= 1` for all real `x`

The slope of the line tangent to the graph of `y=x^3+x` is never less than `1`.