Final answer:
To check the given function for extremum, find the first derivative, set it equal to zero, find the second derivative, and evaluate it at the critical points.
Step-by-step explanation:
To check the given function for extremum, we need to find its critical points. First, we find the first derivative of the function with respect to x and set it equal to zero to find the potential critical x-values. Then, we find the second derivative to determine if these critical points are local extrema.
Step 1: Find the first derivative of u with respect to x
d(u)/dx = 2x + y - 1/x^2 - 1/y^2
Step 2: Set the first derivative equal to zero and solve for x:
2x + y - 1/x^2 - 1/y^2 = 0
Step 3: Find the second derivative of u with respect to x:
d^2(u)/dx^2 = 2 + 2/x^3 + 2/y^3
Step 4: Evaluate the second derivative at the critical points found in step 2:
If the second derivative is positive at a critical point, then that point is a local minimum. If the second derivative is negative, then the critical point is a local maximum. If the second derivative is zero, the test is inconclusive.