Question

f(x,y) = x^2*y-y^2*x+y(x-2)+x

Find all points (x,0) such that the level curve of f at that point is orthogonal to the x-axis

Answer 1

I just wanted to double check my reasoning..

I found the partial derivatives to be:


and I reasoned that to be orthogonal to the x axis, the partial derivative with respect to x would have to be 0.. But plugging y=0 into my partial derivative yields 1 at all values of x.

So did I make a mistake/am I approaching this incorrectly? Or is the answer actually that no points exist..?