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Consider the (nonlinear) mathematical program: max[xy] s.t. x+(1/2)y=8, 0≤x,y.
Without using anything covered in the topic so far, find a global maxi- mum of this problem. Justify your reasoning.
Expert Answer
Ans: X + (1/2) Y = 8
or X = 8 – Y/2 – (i)
Let the function be F(z) = XY
Using (i) we have F(z) = Y(8-Y/2) = 8Y – (Y^2)/2
For finding global maximum, dF(z)/dY = 0
i..e d(8Y – (Y^2)/2)/dY = 0
8 – Y = 0; Hence Y = 8
Outting Y – 8 in (i) we get X = 8 – 8/2 = 4.
Hnece global maximum points are: X = 4; Y = 8 and F(Z) max = max(XY) = 4*8 = 32
