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How do you minimize and maximize #f(x,y)=x-y/(x-y/(x-y))# constrained to #1<yx^2+xy^2<16#?

Answer 1

See below.

This problem can be successfully handled with the Lagrange Multipliers technique.

The local maxima/minima points are

#((f(x,y),x,y, "type"), (-0.750704,-1.71756,1.25284,"maximum"), (1.81974,0.562693,-1.64382,"minimum"), (-3.39363,-4.05723,2.44297,"maximum"), (3.33301,1.4188,-4.14166,"minimum"))#

Attached a plot showing the feasible region superimposed to the objective function level curves, with the local maxima/minima points.

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Answer 2

Cameron Alexander

To minimize and maximize ( f(x,y) = \frac{x - y}{x - \frac{y}{x - \frac{y}{x - y}}} ) subject to the constraint ( 1 < yx^2 + xy^2 < 16 ), you would first find the critical points of ( f(x,y) ) within the given constraint. Then, evaluate the function at those critical points and at the endpoints of the constraint interval to determine the minimum and maximum values of ( f(x,y) ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

How do you minimize and maximize #f(x,y)=x-y/(x-y/(x-y))# constrained to #1&lt;yx^2+xy^2&lt;16#?

How do you minimize and maximize #f(x,y)=x-y/(x-y/(x-y))# constrained to #1<yx^2+xy^2<16#?