Final answer:
The absolute maximum of the given function is 24, and the absolute minimum is -24 on the specified rectangular plate.
Step-by-step explanation:
To find the absolute maximum and minimum of the function f(x, y) = (32x - 8x^2) cos(y) on the rectangular plate defined by 1≤x≤3 and -4π≤y≤4π, we must consider both the interior of the domain and the boundaries.
Since the cosine function oscillates between -1 and 1, the function f(x, y) within the given domain will attain its maximum and minimum values either when cos(y) is 1 or when it's -1. Therefore, to find the extrema of the quadratic in x, 32x - 8x^2, we take its derivative and set it to zero:
64 - 16x = 0 → x = 4. However, since x is bounded by 1 and 3, we check the values of f(x, y) at x = 1 and x = 3. As cos(y) will be either -1 or 1, the extreme values for f(x, y) will occur at the endpoints of x.
For x = 1: f(1, y) = (32−8)cos(y) = 24cos(y). This has a maximum of 24 when cos(y) = 1 and a minimum of -24 when cos(y) = -1.
For x = 3: f(3, y) = (32∓24)cos(y) = 0 for all y since any number times zero is zero.
The absolute maximum of f(x, y) is 24 and the absolute minimum is -24.