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Homogeneous Functions – Homogeneous check to a polynomial multiplication with parameters – Exercise 7043

Exercise

Determine if the following function:

f(x,y)=3x^my^n

Is homogeneous.

Final Answer

 The function is homogeneous of degree m+n

Solution

f(x,y)=3x^my^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)=

=3{(tx)}^m{(ty)}^n=

=3t^mx^mt^ny^n=

=3t^{m+n}x^my^n=

=t^{m+n}f(x,y)

We got

f(tx,ty)=t^{m+n}f(x,y)

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

m+n

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