Final answer:
The question asks to find the absolute maximum and minimum of a function over a specified domain, which involves checking boundary values and finding critical points by using partial derivatives.
Step-by-step explanation:
The student is asking to find the absolute maximum and minimum values of the function f(x, y) = 4x - 6y - x² - y²/6 over the given domain d = (x, y) . To find these values, one must first check the function's values at the boundary of the domain and then find the critical points within the domain by taking the partial derivatives with respect to x and y and setting them equal to zero. After solving for the critical points, evaluate the function at these points and on the boundary. From these values, determine the largest (absolute maximum) and smallest (absolute minimum) values.
To illustrate with a simpler function, suppose we have f(x) = 10 for 0 ≤ x ≤ 20. The graph would be a horizontal line at y=10 from x=0 to x=20. Here the absolute maximum and minimum are both 10, as the function is constant. However, for more complex functions like the one given in the student's question, one must follow the steps mentioned above to find these extrema