12. Find the maximum and minimum of f(x,y) = x² y, subject to the

constraint 4x² + 9y² = 8.

f_x = 2xy = l 8x

f_y = x² = l 18y

So, going to the ratio,

Next,

4x² + 9y² = 4x² +2x² = 6x² = 8Þ x² = 4/3 Þ x = ± 2/Ö 3, y = ± (2/3)^3/2

Finally, we calculate:

Thus, the maximum value, (8/3)(2/3)^3/2 is taken on twice, and the same holds for the minimum value, -(8/3)(2/3)^3/2.