Subtopic: Homogeneous function – Euler's theorem on homogeneous functions
Question Statement:
If $f(x, y)$ is a homogeneous function of degree $n$ having continuous second-order partial derivatives, then $x^2 \frac{\partial^2 f}{\partial x^2} + 2xy \frac{\partial^2 f}{\partial x \partial y} + y^2 \frac{\partial^2 f}{\partial y^2}$ is equal to:
Options:
(A) $n f(x, y)$
(B) $n(n-1) f(x, y)$
(C) $n^2 f(x, y)$
(D) $(n+1) f(x, y)$
Answer Key:
B
Step-by-Step Solution & Explanation
By Euler's extension theorem for homogeneous functions of degree $n$:
$x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = n f$
Differentiating partially with respect to $x$ and $y$ and combining yields:
$x^2 \frac{\partial^2 f}{\partial x^2} + 2xy \frac{\partial^2 f}{\partial x \partial y} + y^2 \frac{\partial^2 f}{\partial y^2} = n(n-1) f(x, y)$
Thus, Option B is correct.