Question

State whether the following statements are true or false Give reasons for your answers.
1. The function f:R^3->R, is given by f(x,y,z)=|x|+|y|+|z| is differentiable at (2,3,-1)
2. The function f(x,y)= max{y/x,X} is a homogeneous function on R^2.
3. The domain of the function f / g where f(x,y)=2xy and g(x,y)=x^2+y^2 is R^2.

Answer 1

1. True. In small neighbourhood of $p=(2,3,-1)$ we have $f(x,y,z)=x+y-z$ that is obviously differentiable in the neighbourhood of $p$.
2. False. Let $a=-1$, then $\max\{ay/x, ax\}\ne a\max\{y/x, x\}$. For example, $x=1,y=7\colon \max \{-7,-1\}\ne-\max\{7,1\}=-7$
3. False. Domain is $\{(x,y)\in\mathbb R^2\colon x^2+y^2>0\}$. Particularly, $\lim\limits_{(x,y)\to(0,0)} \frac{f(x)}{g(x)}$ does not exist because for $(x,y)=(1/n,k/n)$ we have different limits $\lim\limits_{n\to \infty}\frac{f}{g}=\frac{2k}{1+k^2}$ for different $k$