Activity 2.7.3.
Consider the curve defined by the equation \(y(y^2-1)(y-2) = x(x-1)(x-2)\text{,}\) whose graph is pictured in the following figure.
Through implicit differentiation, it can be shown that
\begin{equation*}
\frac{dy}{dx} = \frac{(x-1)(x-2) + x(x-2) + x(x-1)}{(y^2-1)(y-2) + 2y^2(y-2) + y(y^2-1)}\text{.}
\end{equation*}
Use this fact to answer each of the following questions.
(a)
Determine all points \((x,y)\) at which the tangent line to the curve is horizontal. (Use technology appropriately to find the needed zeros of the relevant polynomial function.)
(b)
Determine all points \((x,y)\) at which the tangent line is vertical. (Use technology appropriately to find the needed zeros of the relevant polynomial function.)
(c)
Find the equation of the tangent line to the curve at one of the points where \(x = 1\text{.}\)
