Final answer:
To find the absolute maxima and minima of the function f(x, y) on the specified region, we use partial differentiation and boundary analysis. Critical points are found by setting partial derivatives to zero, and the function is evaluated at these points and along the boundary of the region, including corner points.
Step-by-step explanation:
To find the absolute maxima and minima of the function f(x, y) = x³ - y³ - 27xy on the region R=[1,2] × [3,4], we need to use calculus, specifically through partial differentiation and evaluation at the boundary of the region. We'll look for critical points inside R by setting the partial derivatives with respect to x and y to zero and then check the values of f(x, y) at these critical points. Since the question involves a closed and bounded region, we also need to evaluate the function at the boundaries, which becomes a matter of checking the function along the lines x=1, x=2, y=3, and y=4. Therefore, we must also consider the corner points (1,3), (1,4), (2,3), and (2,4).
We can start by finding the partial derivatives fx and fy and setting them to zero to find any critical points:
fx(x, y) = 3x² - 27y = 0
fy(x, y) = -3y² - 27x = 0
After finding any critical points, we then compare the values at the boundary and at these points to determine the absolute maximum and minimum values.