Question

Let \( f(x, y) \) be a continuously differentiable homogeneous function of degree 4. Which of the following is necessarily true?

• A. \( x \frac{\partial f(x, y)}{\partial x} + y \frac{\partial f(x, y)}{\partial y} = f(x, y) \)
• B. \( x \frac{\partial f(x, y)}{\partial x} + y \frac{\partial f(x, y)}{\partial y} = 2f(x, y) \)
• C. \( x \frac{\partial f(x, y)}{\partial x} + y \frac{\partial f(x, y)}{\partial y} = 4f(x, y) \)
• D. \( x \frac{\partial f(x, y)}{\partial x} + y \frac{\partial f(x, y)}{\partial y} = 8f(x, y) \)

Hint:

For homogeneous functions, use Euler's theorem to relate the function and its partial derivatives. The degree of homogeneity gives the multiplier in the equation.

Answer 1

The Correct Option is C

Step 1: Understanding the concept of homogeneity.
A function \( f(x, y) \) is said to be homogeneous of degree \( n \) if it satisfies the following relation for all \( \lambda \): \[ f(\lambda x, \lambda y) = \lambda^n f(x, y) \] For this question, we are given that the function is homogeneous of degree 4, so we have: \[ f(\lambda x, \lambda y) = \lambda^4 f(x, y) \] Step 2: Applying Euler's theorem for homogeneous functions.
Euler's theorem for homogeneous functions states that for a continuously differentiable homogeneous function of degree \( n \), we have the following relation: \[ x \frac{\partial f(x, y)}{\partial x} + y \frac{\partial f(x, y)}{\partial y} = n f(x, y) \] In this case, since the degree of homogeneity is 4, we get: \[ x \frac{\partial f(x, y)}{\partial x} + y \frac{\partial f(x, y)}{\partial y} = 4 f(x, y) \] Step 3: Conclusion.
Thus, the correct answer is (C) \( x \frac{\partial f(x, y)}{\partial x} + y \frac{\partial f(x, y)}{\partial y} = 4 f(x, y) \).