Find Equations of tangent lines

calculuscurvestangent line

I'm having a hard time figuring this out.
I'm asked to find the the equations of the horizontal lines to the curve of
$$y=x^3-3x+1$$

I set the derivative equal to zero and solve for x, to find the constant of each equation (since they are horizontal), I get $-1$ and $1$. When graphing the function, the $-1$ makes sense graphically, but the $1$ doesn't make sense, I realized that even more when asked to find the equations of the lines perpendicular to the above tangent lines at the point of tangency. When looking for the point of tendency it seems I should be getting $y=3$ instead of $y=1$ (at the top of the bell curve)

I'm completely stuck and wonder what I did wrong.
Thanks a lot for your help and sorry about the formatting, not sure why it didn't work.

prompt

graph

Best Answer

Your way seems correct, starting form

$$f(x)=x^3-3x+1 \implies f'(x)=3x^2-3=0 \implies x=\pm 1$$

and then

  • $f(1)=-1$
  • $f(-1)=3$

and the $2$ horizontal lines are then

$$y=-1, \quad y=3$$

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