Which one of the following functions is not h | vedclass

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(D) function $f(x, y)$ is homogeneous of degree $n$ if $f(\lambda x, \lambda y) = \lambda^n f(x, y)$.$(A)$ $f(\lambda x, \lambda y) = \frac{\lambda x

Vedclass · Accepted Answer

$f(x, y) = x \left[ \ln \frac{2x^2 + y^2}{x} - \ln(x + y) ight] + y^2 	an \frac{x + 2y}{3x - y}$

Vedclass · Answer

(D) function $f(x, y)$ is homogeneous of degree $n$ if $f(\lambda x, \lambda y) = \lambda^n f(x, y)$.$(A)$ $f(\lambda x, \lambda y) = \frac{\lambda x - \lambda y}{(\lambda x)^2 + (\lambda y)^2} = \frac{\lambda(x - y)}{\lambda^2(x^2 + y^2)} = \lambda^{-1} f(x, y)$. This is homogeneous of degree $-1$.$(B)$ $f(\lambda x, \lambda y) = (\lambda x)^{1/3} (\lambda y)^{-2/3} 	an^{-1} \frac{\lambda x}{\lambda y} = \lambda^{1/3 - 2/3} x^{1/3} y^{-2/3} 	an^{-1} \frac{x}{y} = \lambda^{-1/3} f(x, y)$. This is homogeneous of degree $-1/3$.$(C)$ $f(\lambda x, \lambda y) = \lambda x (\ln \sqrt{\lambda^2(x^2 + y^2)} - \ln(\lambda y)) + \lambda y e^{\frac{\lambda x}{\lambda y}} = \lambda x (\ln \lambda + \ln \sqrt{x^2 + y^2} - \ln \lambda - \ln y) + \lambda y e^{x/y} = \lambda f(x, y)$. This is homogeneous of degree $1$.$(D)$ $f(\lambda x, \lambda y) = \lambda x [\ln \frac{2\lambda^2 x^2 + \lambda^2 y^2}{\lambda x} - \ln(\lambda x + \lambda y)] + (\lambda y)^2 	an \frac{\lambda x + 2\lambda y}{3\lambda x - \lambda y} = \lambda x [\ln \frac{\lambda^2(2x^2 + y^2)}{\lambda x} - \ln(\lambda(x + y))] + \lambda^2 y^2 	an \frac{x + 2y}{3x - y} = \lambda x [\ln \lambda + \ln \frac{2x^2 + y^2}{x(x + y)}] + \lambda^2 y^2 	an \frac{x + 2y}{3x - y}$. Since the $\ln \lambda$ term does not factor out as $\lambda^n$,this function is not homogeneous.

Which one of the following functions is not homogeneous?

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