Proof that f(x,y) = 3x² + 4xy + 3y² is a homogeneous function:

A homogeneous function is a function where each term has the same degree. In other words, if we scale the inputs (x, y) by a constant factor λ, the function output scales by the same factor λ^n, where n is the degree of the function.

To prove that f(x, y) = 3x² + 4xy + 3y² is a homogeneous function, we need to show that each term in the function has the same degree.

Step 1: Determine the degree of each term:
The degree of a term is the sum of the exponents of its variables. Let's analyze each term in f(x, y):

- The term 3x² has a degree of 2 (x raised to the power of 2).
- The term 4xy has a degree of 2 (x raised to the power of 1 and y raised to the power of 1).
- The term 3y² has a degree of 2 (y raised to the power of 2).

Step 2: Compare the degrees of the terms:
Since each term in f(x, y) has a degree of 2, we can conclude that the function is a homogeneous function.

Checking Euler's theorem:

Euler's theorem for homogeneous functions states that if f(x, y) is a homogeneous function of degree n, then:

x * ∂f/∂x + y * ∂f/∂y = n * f(x, y)

Let's apply Euler's theorem to f(x, y) = 3x² + 4xy + 3y²:

- ∂f/∂x = 6x + 4y
- ∂f/∂y = 4x + 6y

Now, let's check if Euler's theorem holds for f(x, y):

x * ∂f/∂x + y * ∂f/∂y = x(6x + 4y) + y(4x + 6y)
= 6x² + 4xy + 4xy + 6y²
= 10x² + 8xy + 6y²

n * f(x, y) = 2(3x² + 4xy + 3y²)
= 6x² + 8xy + 6y²

Since x * ∂f/∂x + y * ∂f/∂y = n * f(x, y), Euler's theorem holds true for f(x, y) = 3x² + 4xy + 3y².

Therefore, we have proven that f(x, y) = 3x² + 4xy + 3y² is a homogeneous function of degree 2 and Euler's theorem holds for this function.