Question

For \[F(x,y)=\frac{y\cos\left(\frac{y}{x}\right)+x}{x\cos\left(\frac{y}{x}\right)}\], what is \[F(\lambda x,\lambda y)\]?

Options

• \[\frac{y\cos\left(\frac{y}{x}\right)+x}{x\cos\left(\frac{y}{x}\right)}=F(x,y)\]

• \[\frac{1}{\lambda}\frac{y\cos\left(\frac{y}{x}\right)+x}{x\cos\left(\frac{y}{x}\right)}\]

• \[\lambda^{2}\frac{y\cos\left(\frac{y}{x}\right)+x}{x\cos\left(\frac{y}{x}\right)}\]

• \[\lambda\frac{y\cos\left(\frac{y}{x}\right)+x}{x\cos\left(\frac{y}{x}\right)}\]

Answer 1

The ratio \[\frac{\lambda y}{\lambda x}\] equals \[\frac{y}{x}\], while \[\lambda\] factors cancel from numerator and denominator. Hence \[F(\lambda x,\lambda y)=F(x,y)\], so \[F(x,y)\] is homogeneous of degree \[0\].