Final answer:
To find the absolute maximum and minimum values of the function on the quarter disc, we must analyze the function on the boundary and at any critical points, then compare these values.
Step-by-step explanation:
To find the absolute maximum and minimum values of the function f(x, y) = x + y + √(1 - x^2 - y^2) on the quarter disc where x ≥ 0, y ≥ 0, and x^2 + y^2 ≤ 1, we first need to consider that the function is defined on a closed and bounded set. Therefore, by the Extreme Value Theorem, the function f has both an absolute maximum and an absolute minimum on this domain.
We can find these extrema by analyzing the function on the boundary as well as at any critical points within the quarter disc. The boundary of the quarter disc includes the positive x-axis, positive y-axis, and the quarter of the unit circle where both x and y are nonnegative. Moreover, the critical points are found where the gradient of f is zero or undefined, inside the quarter disc.
Once we have evaluated f at these points and along the boundaries, we compare these values to determine the absolute maximum and absolute minimum values.