$$B=\{x(\alpha-x)~|~x\in(0,1)\},$$ where $~\alpha \in \mathbb Q $ is a fixed parameter
My attempt :
- $\alpha=0$
$$x(\alpha-x)=-x^2 $$ $$ \inf(B)=-1$$
- $\alpha<0$ $$x(\alpha-x)\le0 $$ $$ \inf(B)=\alpha -1$$
- $\alpha\ge1$ $$x(\alpha-x)\ge x $$ $$\inf(B)=0$$
- $\alpha \in(0,1) $ $$x(\alpha-x),~~\alpha-1<0 $$ $$ \inf(B)=\alpha-1$$