Mathematics

Problem

The curve defined by

x^{3}+y^{3}-9xy=0

is known as the folium of Descartes. (a) Compute

\dfrac{dy}{dx}

for the curve in terms of

x

and

y

. (b) Determine all ordered pairs

(x,y)

on the curve for which the tangent line is horizontal. (c) For each point found in part (b), verify that the tangent line is not vertical by checking the denominator of

\dfrac{dy}{dx}

. Provide justification. (d) Write the equation of every horizontal tangent line to the curve.

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