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

Show final answer

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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