Find points on a curve when slope is 0

[MrJamesta]

Homework Statement
Find the points on the curve xy^2+x^2y=16 when the tangent line is horizontal

The attempt at a solution
I found the derivative of the curve
-(y(2x+y))/(x(x+2y))
then I found what values of y make the derivative equal 0
y=0,-2x
Then I went to plug into the original curve to find x and that is where I got lost. I do not know how to isolate x in this case.

 

[LCKurtz]

Homework Statement
Find the points on the curve xy^2+x^2y=16 when the tangent line is horizontal

The attempt at a solution
I found the derivative of the curve
-(y(2x+y))/(x(x+2y))
then I found what values of y make the derivative equal 0
y=0,-2x
Then I went to plug into the original curve to find x and that is where I got lost. I do not know how to isolate x in this case.

The first thing you should notice is that $y=0$ doesn't give a point on the graph. But what happens if you put $y=-2x$ into the original equation?

 

[MrJamesta]

Thanks! Got it now.