Allied & ICAR Exam Bank • ALLIED_0026 NAT • 2 Marks

Section 1: Engineering Mathematics

Calculus

Subtopic: Homogeneous function – Euler's theorem on homogeneous functions

Question Statement:

If $f(x, y) = x^4 + x^2 y^2 + y^4$, then $x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y}$ is equal to $k f(x, y)$. The value of $k$ is:

Answer Key:

4

Step-by-Step Solution & Explanation

Function $f(x, y)$ is homogeneous of degree $n = 4$. By Euler's theorem, $x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = n f(x, y) = 4 f(x, y)$.
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