   Chapter 2.6, Problem 41E

Chapter
Section
Textbook Problem

# Fanciful shapes can be created by using the implicit plotting capabilities of computer algebra systems.(a) Graph the curve with equation y ( y 2 − 1 ) ( 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).

To determine

(a)

To graph:

The curve and find how many point have horizontal tangents and what are the x –coordinates of these points:

Explanation

The graph of given curve is

To determine

(b)

To find:

Equation of tangent lines at points (0, 1) and (0, 2)

To determine

(c)

To find:

The exact x- coordinates of points in part a)

To determine

(d)

To create:

More fanciful curves by modifying the equation

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