Homogeneous Functions – Homogeneous check to function multiplication with ln – Exercise 7034

Exercise

Determine if the following function:

$f(x,y)=xy\ln\frac{x+3y}{4x+y}$

Is homogeneous.

Final Answer

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Solution

$f(x,y)=xy\ln\frac{x+3y}{4x+y}$

Function f is called homogeneous of degree r if it satisfies the equation:

$f(tx,ty)=t^nf(x,y)$

for all t.

$f(tx,ty)=$

$=tx\cdot ty\ln\frac{tx+3ty}{4tx+ty}=$

$=t^2xy\ln\frac{t(x+3y)}{t(4x+y)}=$

$=t^2xy\ln\frac{x+3y}{4x+y}=$

$=t^2f(x,y)$

We got

$f(tx,ty)=t^2f(x,y)$

Hence, by definition, the given function is homogeneous of degree 2.

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