Find all absolute minimums and maximums of the function f (x, y) = x2 + 2xy + y2

Question

Find all absolute minimums and maximums of the function f (x, y) = x2 + 2xy + y2 over the domain D = {(x, y) : x2 + y2 ? 8} (i.e. the closed disk of radius ?8 centered at (0, 0)). Note: this is a continuous function defined on a closed and bounded domain, so by the Extreme Value Theorem there must be at least one point in the domain which is an absolute maximum and a point in the domain which is an absolute minimum. Hint: You can use Langrange multipliers to check for the candidate extreme values on the boundary of the domain, and there may be many critical points in the interior of D.

Explanation / Answer

df/dx=0 2x+2y=0 df/dy=0 2x+2y=0 x=-y x^2+y^2=8 2x^2=8 x=2,-2 y=-2,2

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