Consider the function fx1 x2 x13 x23 3x1 x22 Verify that

Consider the function f(x_1, x_2) = x_1^3 + x_2^3 + 3x_1 x_2^2. Verify that f() is homogeneous, by applying Equation 2.121 on NS p. 55. Find the degree, k, of which this function is homogeneous. Use f() to verify Euler\'s equation by directly applying equation 2.123 on NS p. 56.

Solution

F(x1,x2) = x1^3 + x2^3+3x1x2^2

To chek the homogeneity we replace all variables by k*variable

F(x1,x2) = (kx1)^3 + (kx2)^3+3kx1(kx2)^2

=K^3* x1^3 + k^3*x2^3 + k^3*3x1x2^2

=K^3(x1^3 + x2^3+3x1x2^2)

=K^3 F(x1,x2)