Final answer:
To find the absolute maxima and minima of the function on the unit disk, we identify critical points inside the disk and extreme values on the boundary, then compare the values to determine the absolute maximum and minimum.
Step-by-step explanation:
To find the absolute maximum and minimum values of the function f(x, y) = 4x³ + 3y² on the unit disk D = (x² + y² ≤ 1), we need to consider both the interior and the boundary of D. Inside the disk, we find the critical points by taking the gradient of f and setting it to zero which means simultaneously solving ∂f/∂x = 0 and ∂f/∂y = 0.
For the boundary, we apply the method of Lagrange multipliers or parameterize the boundary and substitute it into the function to find the extreme values. Once we have the list of critical points and extreme values on the boundary, we compare their function values to identify the absolute maximum and minimum.
The unit disk is a common domain for optimization problems in two variables, and the method combines calculus with algebraic manipulation.