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

Answer 1

# f(x,y) # is a homogeneous function of degree 3

# f(x,y) = -x^3 - 4xy^2 + y^3 #
Each term in # f(x,y) # 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(x,y) # is a homogeneous function of degree 3
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Answer 2

See below.

A Homogeneous function, is a function which obeys the relationship

#f(lambda x_1,lambda x_2,cdots,lambda x_n)=lambda^nf(x_1,x_2,cdots, x_n)#

In the present case we have

#f(lambda x, lambda y) = -(lambda x)^3-2(lambdax)(lambda y)^2+(lambday)^3=lambda^3(-x^3-xy^2+y^3)=lambda^3f(x,y)#
so #f(x,y)# is homogeneous with degree #3#
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Answer 3

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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Answer from HIX Tutor

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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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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