Absolute Maxima and Minima on a Closed and Bounded Region

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Consider the function \[f(x,y) = 1 + x + y - \dfrac{1}{2}x^2 - \dfrac{1}{2} y^2.\] We wish to find the absolute extrema of \(f(x,y)\) in the triangular region with vertices \((0,0)\), \((4,0)\) and \((0,4)\).

The following shows the critical points of \(f(x,y)\) both in the interior and on the boundary of the region as well as the endpoints of the boundary. The maxima and minima may occur at any of these points.

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Joseph Lo, Created with GeoGebra