I've been trying to figure out how can I solve this exercise but I haven't had much luck so far. Do you think you can help me out a bit? Pointing out what might I possibly be doing wrong?
The exercise is as follows:
Find the coordinates where the tangent to the curve is horizontal
$x^3+3xy+2y^2+4y=1$
Given that it's difficult to solve for either x or y. I decided to differentiate implicitly. And here's what I got:
$- {3x^2+3y\over 4y+3x+4}=0 $
In order to find the horizontal tangents, the first order differential must be zero, and for this case particularly:
$ 3x^2+3y=0 $
Now, solving for x:
$x=\sqrt{-y}$ $x=-\sqrt{-y}$
Which tells me that y must be positive. (Real field)
But now I'm stuck there. Just looking at the answers I can't think of anything else but some numbers that might satisfy the equation; $(1,-1)$, $(-1,-1)$,$(0,0)$ But I wouldn't know how to get there, nor I know if those are the right coordinates. Can you help me out? Thanks in advance.