Find a point (if exists), Besides (0,0), where the graph of xy = x + y^2 has a nonexistence slope (also known as vertical tangent). (Use calculus to show if there is another point out there)

Question

I have a calculus test tomorrow, please help!!

thank you

Answer 1

xy = x + y²

y + x(dy/dx) = 1 + 2y(dy/dx)

(dy/dx)(x-2y) = 1-y

dy/dx = (1-y) / (x-2y)

Vertical tangent when x-2y = 0 (so, x = 2y)

But, xy = x + y². So, 2y² = 2y + y²

y² - 2y = 0

y(y - 2) = 0

y = 0 or y = 2

If y = 0, then x = 2y = 0

If y = 2, then x = 2y = 4

Vertical tangent at (0,0) and at (4,2).