Consider the function f(x, y) = x2 + xy + y2 defined on the unit disc, namely, d = {(x, y)| x2 + y2 ≤ 1}. use the method of lagrange multipliers to locate the maximum and minimum points for f on the unit circle. use this to determine the absolute maximum and minimum values for f on d. maximum minimum

First we note that the partial derivatives vanish simultaneously at one point:



so there is one critical point within the region .

The Lagrangian is



and has partial derivatives





Set each partial derivative to 0 to find the possible critical points within the disk . Then we notice that







Since , we have



And since , or , we also have



So we have four possible additional critical points to consider:







It should be clear enough which of these correspond to the absolute extrema of over .

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