Exercise
Determine if the following function:
f(x,y)=x^m+x^{m-n}y^n
Is homogeneous.
(m,n parameters)
Final Answer
f(x,y)=x^m+x^{m-n}y^n
Function f is called homogeneous of degree r if it satisfies the equation:
f(tx,ty)=t^rf(x,y)
for all t.
f(tx,ty)=
=(tx)^m+(tx)^{m-n}\cdot (ty)^n=
=t^m\cdot x^m+t^{m-n}\cdot x^{m-n}\cdot t^n\cdot y^n=
=t^m\cdot x^m+t^m\cdot x^{m-n}\cdot y^n=
=t^m(x^m+x^{m-n}\cdot y^n)=
=t^mf(x,y)
We got
f(tx,ty)=t^mf(x,y)
Hence, by definition, the given function is homogeneous of degree m.
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