Find the absolute maximum and minimum of values of the set d. f(x, y)= 2x^2+y^2-4x-2y+1 d={(x, y) | 0 < = 3, 0 < = y < = 2}

Find the critical points within the region:




So (1, 1) is the only critical point of the function and it happens to fall within the rectangle . At this point, we get a value of .

Now check along the boundaries for potential extreme values.

If , then , which has a maximum when and has a minimum when . So we have a potential extrema at and .

If , then , with a max when and min when . So we have and .

If , then , with a max when and a min when . So we have and .

If , then again, with a max when and a min when . So we have, again, and .

So there are absolute maxima of 7 at (3, 0) and (3,2), and an absolute minimum of -2 at (1, 1).