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

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See below.

A Homogeneous function, is a function which obeys the relationship

In the present case we have

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To determine if a function ( f(x, y) = -x^3 - 4xy^2 + y^3 ) is homogeneous, we check if it satisfies the condition:

[ f(tx, ty) = t^n f(x, y) ]

where ( t ) is a scalar and ( n ) is the degree of homogeneity.

For ( f(x, y) = -x^3 - 4xy^2 + y^3 ): [ f(tx, ty) = -(tx)^3 - 4(tx)(ty)^2 + (ty)^3 = -t^3x^3 - 4t^3x y^2 + t^3y^3 ]

Now, we compare this with ( t^n f(x, y) ): [ t^3 f(x, y) = t^3(-x^3 - 4xy^2 + y^3) = -t^3x^3 - 4t^3xy^2 + t^3y^3 ]

Since ( f(tx, ty) = t^3 f(x, y) ), the function is homogeneous of degree ( n = 3 ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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