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Old Mar 11th 2017, 01:26 AM   #1
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Join Date: Mar 2017
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Extrema of two variable function

The temperature in a rectangular plate bounded by the lines x=0, y=0, x=3 and y=5 is T=xy^2-x^2y+100. What is the maximum temperature difference between two points on the plate?
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Old Mar 11th 2017, 04:52 AM   #2
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Extrema of two variable function

You can take the differential with respect to X and Y to find that the extremum sit on the line y = X/2 and for the minimum and Y = 2x for the maximum. These increase with X and Y so the maximum sits on the point 5,2.5 and the minimum on the point 1.5,3

They will be 131.25 and 93.25 respectively giving a maximum temparature difference of 38.

PS. There might also be a clever way to formulate this problem as a lagrange multiplier, but I am not really sure if it is possible because of the complexity of setting the constraints to zero.
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