Step 1

The domain is the whole (x, y) plane because the denominator is always positive.

Step 2

= -[2(x + y) + 2(x + l)][(x + y)² + (x + 1)² + y²] ^-2, = -[2(x + y) + 2y][(x + y)² + (x + 1)² + y²] ^-2.

Step 3

The partial derivatives are zero when

2(x + y) + 2(x + 1) = 0, 2(x + y) + 2y = 0,

or

2x + y + 1 = 0, x + 2y = 0.

The critical point is

x = -⅔, y = ⅓, and z = 3.

Step 4

Let E be the hyperreal region -H ≤ x ≤ H, -H ≤ y ≤ H where H is positive infinite.

Step 5

At a boundary point of E where x = ±H, (x + 1)² is infinite so z is infinitesimal. At a boundary point where y = ±H, y² is infinite so again z is infinitesimal.