Question

Let \( f(x, y, z) = x^2 y^3 z \). Then, what is the value of \[ x \frac{\partial f}{\partial x}(x, y, z) + y \frac{\partial f}{\partial y}(x, y, z) + z \frac{\partial f}{\partial z}(x, y, z)? \]

• A. \( f(x, y, z) \)
• B. \( 2f(x, y, z) \)
• C. \( 3f(x, y, z) \)
• D. \( 6f(x, y, z) \)

Hint:

Euler's Theorem for homogeneous functions states that for a function \( f(x, y, z) \) that is homogeneous of degree \( n \), \[ x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} + z \frac{\partial f}{\partial z} = n f(x, y, z) \] Here, \( f \) is of degree 6 (2 from \( x \), 3 from \( y \), and 1 from \( z \)).

Answer 1

The Correct Option is D

Let us first compute the partial derivatives of the function: \[ f(x, y, z) = x^2 y^3 z \] \[ \frac{\partial f}{\partial x} = 2x y^3 z,\quad \frac{\partial f}{\partial y} = 3x^2 y^2 z,\quad \frac{\partial f}{\partial z} = x^2 y^3 \] Now, multiply each term with its respective variable: \[ x \cdot \frac{\partial f}{\partial x} = 2x^2 y^3 z,\quad y \cdot \frac{\partial f}{\partial y} = 3x^2 y^3 z,\quad z \cdot \frac{\partial f}{\partial z} = x^2 y^3 z \] Add all the terms: \[ 2x^2 y^3 z + 3x^2 y^3 z + x^2 y^3 z = 6x^2 y^3 z = 6f(x, y, z) \]