Problem 41E

41. Fanciful shapes can be created by using the implicit plotting capabilities of computer algebra systems.

(a) Graph the curve with equation

$$

y\left(y^{2}-1\right)(y-2)=x(x-1)(x-2)

$$

At how many points does this curve have horizontal tangents? Estimate the $x$ -coordinates of these points.

(b) Find equations of the tangent lines at the points $(0,1)$ and $(0,2)$.

(c) Find the exact $x$ -coordinates of the points in part (a).

(d) Create even more fanciful curves by modifying the equation in part (a).

 

Step-by-Step Solution