i have a doubt with this two exercises:
a) $g(x,y)=\sqrt{x+y}(4x+3y)$ is homogeneous?
Let $\lambda\in \mathbb{R}$ then $g(\lambda x,\lambda y)=\sqrt{\lambda x+\lambda y}(4\lambda x+3\lambda y)=\sqrt{\lambda(x+y)}(\lambda(4x+3y))=\lambda^{\frac{3}{2}}\sqrt{x+y}(4x+3y)=\lambda^{\frac{3}{2}}g(x,y)$ then $g$ is homogeneous of grade $\frac{3}{2}$. Is correct this?.
b) $f(x,y)=\frac{\ln(x^3)}{\ln(y^3)}$
Let $\lambda\in \mathbb{R}$ then $f(\lambda x, \lambda y)=\frac{\ln(\lambda^3x^3)}{\ln(\lambda^3y^3)}=\frac{\ln(\lambda^3)+\ln(x^3)}{\ln(\lambda^3)+\ln(y^3)}\not =\lambda^nf(x,y)$ then $f$ is not homogeneous.
I have serious doubt in this two exercise, can someone help me?
Yes it is correct by the definition of homogeneous function.