Find the smallest value of x for which the following curve has vertical tangent lines: y= x^2 + xy^2 - x/ x^2

Answer 1

since it's kind of dumb just to have x/x^2, I assume you meant
y = (x^2 + xy^2 - x)/x^2
y = (x+y^2-1)/x
xy = x+y^2-1
xy - y^2 - x + 1 = 0
This is a degenerate conic -- just two straight lines:
y=1 and y=x-1
So, why don't use use some parentheses and tell us what you really meant ...