Exercise
Determine if the following function:
f(x,y)=xy\ln\frac{x+3y}{4x+y}
Is homogeneous.
Final Answer
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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