Question

A function $F(x,y)$ is a homogeneous function of degree $n$, if $F(x,y)=x^{n}g\left(\frac{y}{x}\right)$ or $y^{n}h\left(\frac{x}{y}\right)$
On the basis of above information, which of the following function is homogeneous function of degree 0 ?

• A. $F_1(x,y)=y^2+2xy$
• B. $F_2(x,y)=2x-3y$
• C. $F_3(x,y)=\cos \left(\frac{y}{x}\right)$
• D. $F_4(x,y)=\sin x+\cos y$

Answer 1

Answer: (C)

A function $F(x,y)$ is a homogeneous function of degree $n$ if
$F(x,y)=x^{n}g\left(\frac{y}{x}\right)\text{ or }{y}^{n}h\left(\frac{x}{y}\right)$
We also observe that
$F_1(x,y)=x^2\left(\frac{y^2}{x^2}+\frac{2y}{x}\right)=x^2{h}_1\left(\frac{y}{x}\right)$
or $F_1(x,y)=y^2\left(1+\frac{2x}{y}\right)=y^2{h}_2\left(\frac{x}{y}\right)$
$F_2(x,y)=x\left(2-\frac{3y}{x}\right)=x{h}_3\left(\frac{y}{x}\right)$
or $F_2(x,y)=y\left(2\frac{x}{y}-3\right)=y{h}_4\left(\frac{x}{y}\right)$
$F_3(x,y)=x^0\cos \left(\frac{y}{x}\right)=x^0{h}_5\left(\frac{y}{x}\right)$
$F_4(x,y)\ne x^n{h}_6\left(\frac{y}{x}\right)$
for any $n\in W$
or $F_4(x,y)\ne y^n{h}_7\left(\frac{x}{y}\right),$ for any $n\in W$
$\because F_3(x,y)=x^0{h}_5\left(\frac{y}{x}\right)$
$\Rightarrow F_3(x,y)$ is a homogeneous function of degree 0 .