What is the graphic for $f (x, y) = x y^{\frac{x}{y}} = C_{0}$ $f(x,y)=x y^(x/y)=C_0$ ?

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What is the graphic for $f (x, y) = x y^{\frac{x}{y}} = C_{0}$ $f(x,y)=x y^(x/y)=C_0$ ?

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Answer 1 · 2016-11-04T10:28:36

See below.

Explanation:

Making $x = λ y$ $x = lambda y$ and substituting in

$f (x, y) = x y^{\frac{x}{y}} = c_{0}$ $f(x,y)=x y^(x/y)=c_0$ we have

$f (λ y, y) = λ y y^{λ} = λ y^{λ + 1} = c_{0}$ $f(lambda y,y)=lambda y y^lambda=lambda y^(lambda+1)=c_0$

so

$y (λ) = {(\frac{c_{0}}{λ})}^{\frac{1}{λ + 1}}$ $y(lambda)=(c_0/lambda)^(1/(lambda+1))$ and also

$x (λ) = λ {(\frac{c_{0}}{λ})}^{\frac{1}{λ + 1}}$ $x(lambda)=lambda (c_0/lambda)^(1/(lambda+1))$ so we can generate $x, y$ $x,y$ pairs obeying the condition.

$f (x, y) = x y^{\frac{x}{y}} = c_{0}$ $f(x,y)=x y^(x/y)=c_0$

Attached a plot of

$x (λ), y (λ)$ $x(lambda),y(lambda)$ for $λ \in [0.1, 3]$ $lambda in [0.1,3]$

enter image source here

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What is the graphic for $f (x, y) = x y^{\frac{x}{y}} = C_{0}$ $f(x,y)=x y^(x/y)=C_0$ ?

1 Answer

Explanation:

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What is the graphic for f(x,y)=x y^(x/y)=C_0f(x,y)=xyxy=C0f(x,y)=x y^(x/y)=C_0 ?

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Explanation:

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What is the graphic for $f (x, y) = x y^{\frac{x}{y}} = C_{0}$ $f(x,y)=x y^(x/y)=C_0$ ?