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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

 The function is homogeneous of degree 2

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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