Final answer:
To find the absolute maximum and minimum values of the function f(x, y) = xy² on the set d, we need to evaluate the function within the given constraints and check for critical points. By taking the partial derivatives of f and setting them equal to zero, we find that the only valid critical point is (0, 0). However, this point is not within the set d. To find the absolute maximum and minimum values, we need to check the boundary of set d, which is x²y² = 3. Substituting this condition into f(x, y), we get the absolute maximum value as negative infinity and the absolute minimum value as 0.
Step-by-step explanation:
To find the absolute maximum and minimum values of f on the set d, we need to evaluate the function f(x, y) = xy² within the given constraints. The set d is defined as (x, y) where x ≥ 0, y ≥ 0, and x²y² ≤ 3.
First, let's find the critical points by taking the partial derivatives of f and setting them equal to zero:
- ∂f/∂x = y² = 0, which gives us x = 0 and y = 0.
- ∂f/∂y = 2xy = 0, which gives us x = 0 and y = 0.
However, (0, 0) is not within the constraints of set d, so it is not a valid critical point. To find the absolute maximum and minimum values, we need to check the boundary of the set d. The only boundary is x²y² = 3, which means x²y² is equal to 3. Now, substitute this condition into the function f, we get f(x, y) = xy² = 3/x². Since x² > 0, f(x, y) will have a minimum value as x approaches positive infinity and a maximum value as x approaches negative infinity. Therefore, the absolute maximum value of f is negative infinity and the absolute minimum value is 3/∞ which is equal to 0.