Question

Find the coordinates of the point where the folium has a vertical tangent.

Folium of Descartes, x^3+y^3-9xy=0

Answer 1

\[3x^2 + 3y^2 \frac{dy}{dx} - 9(x \frac{dy}{dx} + y) =0\]

Answer 2

\[\frac{dy}{dx}(3y^2 -9x) = (9y-3x^2) \]

Answer 3

\[\frac{dy}{dx}= \frac{ 9y-3x^2}{3y^2-9x}\]

Answer 4

This derivative is undefined when denominator equals zero.

Answer 5

\[(y \sqrt{3} - 3\sqrt{x})(y \sqrt{3}+3\sqrt{x}) =0\]

Answer 6

\[y= +- \sqrt{3x}\]

Answer 7

In terms of substituting that condition into the given equation above. mmmm that will be difficult to find an exact solution for x.