finding the number of critical points of an implicitly defined equation

Question

How do you find the number of critical points of an implicit equation such as $xy(x-6y)=9a^3$ ?

I have managed to differentiate and get $$\frac{dy}{dx} = 6y^2 - 2xy.$$ I don't know if I'm on the right route.

Answer 1

You can use the implicit derivative: $$2xy+x^2y'-6y^2-6x\cdot 2yy'=0$$

assuming that $$y=y(x)$$ or you get the explicit function:

$$y_{1,2}=\frac{1}{12}x\pm\sqrt{\frac{1}{144}x^2-\frac{3}{2}a^3}$$ and compute the derivative directly