Final answer:
To find the absolute maximum and minimum values of the function f(x, y) = xy² + 7 on the quarter circle domain with a radius √3, we must evaluate the function at both the critical points within the domain and along its circular boundary.
Step-by-step explanation:
The function in question is f(x, y) = xy² + 7, and it's defined over the domain D, where x ≥ 0, y ≥ 0, and x² + y² ≤ 3. This domain represents a quarter circle in the first quadrant (since both x and y are non-negative) with a radius √3. To find the absolute maximum and minimum values of f on D, we must check the boundary of D, which is the circle x² + y² = 3, as well as the interior of D. Since the function is differentiable, we will use the method of Lagrange multipliers to find the critical points on the boundary. For the interior, we calculate the gradient of f and set it equal to zero to find critical points.
After determining the critical points, we evaluate the function f at these points and at the boundary (using the parameterization of the circular boundary, for example x = √3cos(θ), y = √3sin(θ)) to find the maximum and minimum function values. The absolute maximum and minimum will be among these values, and we must compare them to find which is the largest and smallest.