Locating Horizontal Tangent Lines

For $$f(x)=x^3-3*x+1$$, setting the function equal to zero instead of its derivative leads to an incorrect identification of horizontal tangents.

For $$f(x)=x^3-3*x+1$$, the derivative is $$f'(x)=3*x^2-3$$; setting this equal to zero gives $$3*x^2-3=0 \Rightarrow x^2=1\Rightarrow x=1 \text{ or } x=-1$$, indicating horizontal tangent lines at these x-values.

Computing the derivative of $$f(x)=x^3-3*x+1$$ and finding values where it is nonzero does not help in locating horizontal tangent lines.

Using the second derivative of $$f(x)=x^3-3*x+1$$ rather than the first derivative is the proper method for finding horizontal tangents.