The function \(F(x, y) = \displaystyle\frac{x^4 + x^2y^2}{x + y}\) is a homogeneous function of degree
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 3
A function F(x, y) is homogeneous of degree n if \(F(\lambda x, \lambda y) = \lambda^n F(x, y)\) for every non-zero \(\lambda\).
Replacing x by \(\lambda x\) and y by \(\lambda y\):
\( \displaystyle F(\lambda x, \lambda y) = \frac{\lambda^4 x^4 + \lambda^4 x^2y^2}{\lambda x + \lambda y} \)
\( \displaystyle F(\lambda x, \lambda y) = \frac{\lambda^4 (x^4 + x^2y^2)}{\lambda (x + y)} \)
\( F(\lambda x, \lambda y) = \lambda^3 F(x, y) \)
So the degree is 3: the numerator has degree 4 and the denominator degree 1.