Operations Research by Kanti Swarup & P.K Gupta

Operations Research by Kanti Swarup & P.K Gupta

 

0$ and $x \in \mathbb{R}^3$. Then which of the following statements are true. 1. If $f(x)=f(y)$ whenever $\|x\|=\|y\|=\beta$ for a $\beta>0$, then $f(x)=\beta|| x \|^\alpha$ 2. If $f(x)=f(y)$ whenever $\|x\|=\|y\|=1$ for a $\beta>0$, then $f(x)=\|x\|^\alpha$ 3. If $f(x)=f(y)$ whenever $\|x\|=\|y\|=\beta$ for a $\beta>0$, then $f(x)=c\|x\|^\alpha$ for a some constant c. 4. If $f(x)=f(y)$ whenever $\|x\|=\|y\|$, then $f$ must be constant function.