# How do you determine if f(x,y)=-x^3-4xy^2+y^3 is homogeneous and what would it's degree be?

Nov 8, 2016

$f \left(x , y\right)$ is a homogeneous function of degree 3

#### Explanation:

$f \left(x , y\right) = - {x}^{3} - 4 x {y}^{2} + {y}^{3}$

Each term in $f \left(x , y\right)$ is a power of $x$ and $y$ only, and the sum of the powers of each term is the same (in this case 3). This meets the definition of a homogeneous function.

Hence $f \left(x , y\right)$ is a homogeneous function of degree 3

Nov 8, 2016

See below.

#### Explanation:

A Homogeneous function, is a function which obeys the relationship

$f \left(\lambda {x}_{1} , \lambda {x}_{2} , \cdots , \lambda {x}_{n}\right) = {\lambda}^{n} f \left({x}_{1} , {x}_{2} , \cdots , {x}_{n}\right)$

In the present case we have

$f \left(\lambda x , \lambda y\right) = - {\left(\lambda x\right)}^{3} - 2 \left(\lambda x\right) {\left(\lambda y\right)}^{2} + {\left(\lambda y\right)}^{3} = {\lambda}^{3} \left(- {x}^{3} - x {y}^{2} + {y}^{3}\right) = {\lambda}^{3} f \left(x , y\right)$

so $f \left(x , y\right)$ is homogeneous with degree $3$