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Absolute Minimum and Maximum Value of a Function on a Set

Find the absolute maximum and minimum values of f on the set D.
28. f(x, y) = 3 + xy ? x ? 2y, D is the closed triangular region with vertices (1, 0), (5, 0), and (1, 4)
24. Use Lagrange multipliers to prove that the triangle with maximum area that has a given perimeter p is equilateral.


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From the matrix of 2nd derivatives of f(x,y) we see that is does not produce a sign-defined quadratic form, therefore f(x,y) has no local extrema anywhere, and so its absolute extrema in a bounded region ar on the boundaries of the region.
In this case these are the sides of the triangle.
The side (1,0)-(5,0) has y = 0 and so f = 3 - x on it, with extrema at the vertices (1,0) and (5,0), equal to 2 and -2, respectively.
The side (1,0)-(1,4) has x = 1 and so f = 2 - ...

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