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

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

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Answer 1 · 2017-02-20T12:58:55

See below.

Explanation:

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.

enter image source here

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

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How do you minimize and maximize f(x,y)=x-y/(x-y/(x-y))f(x,y)=x−yx−yx−yf(x,y)=x-y/(x-y/(x-y)) constrained to 1<yx^2+xy^2<161<yx2+xy2<161<yx^2+xy^2<16?

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Explanation:

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