Show that (0, 0) is the only critical point of f(x,y)=(xy)/(x^2+y^2+1)

Question

I know that to find the critical point, you take the partial derivatives with respect to x and y and find the values of x and y when the partial derivatives are equal to zero. I set the partial derivatives equal to each other to find when they both equal zero at the same point. I am having trouble with the algebra from here.

Answer 1

f(x,y) = xy/(x² + y² +1)

∂f/∂x = y(y² - x² + 1)/(x² + y² + 1)²

∂f/∂y = x(x² - y² + 1)/(x² + y² + 1)²

Critical points occurs where the function is not differentiable, or where both

∂f/∂x = 0 and

∂f/∂y = 0

The only possibility for f(x,y) not to be differentiable is where the denominator is 0.

But for all x,y we have x² + y² + 1 > 0.

Therefore from now on we can ignore the denominator and look for those x and y satisfying both

y(y² - x² + 1) = 0

x(x² - y² + 1) = 0

Case 1: x = y = 0

That is the point mentioned in the problem statement

Case 2: x = 0, y ≠ 0

y(y² - x² + 1) = y(y² + 1)

x(x² - y² + 1) = 0

In this case y(y² + 1) ≠ 0 because y ≠ 0 and y² + 1 > 0

Case 3: x ≠ 0, y = 0

Symmetrical to Case 2

Case 4: x ≠ 0, y ≠ 0

Then we need to find x and y satisfying both

y² - x² + 1 = 0

x² - y² + 1 = 0

Case 4.1 x² ≤ y²

Then y² - x² + 1 > 0

Case 4.2 y² ≤ x²

Then x² - y² + 1 > 0

Consequently only Case 1 yields a critical value.