Question

Which of the following is not a homogeneous function?

• A. \(y^{2}+2xy\)
• B. \(2x-3y\)
• C. \(\mathrm{sin}(\frac{y}{x})\)
• D. \(\mathrm{cos}x+\mathrm{sin}y\)

Hint:

A function is homogeneous if \(f(\lambda x,\lambda y)=\lambda^nf(x,y)\) for some n.

Answer 1

The Correct Option is D

Step 1: Idea
Every term of a homogeneous function must scale by the same power of lambda.

Step 2: Check D
The terms $\cos x$ and $\sin y$ change in a non-power way, for example $\cos(\lambda x)\ne\lambda^n\cos x$.

Step 3: Result
So option (D) is the one that is not homogeneous.

Final Answer:
Only cos x + sin y fails the scaling test. \[ \boxed{\text{(D) }\cos x+\sin y} \]