## Determine $\sup \{xy−x^2/2\mid x\in [-1,1]\}$

I am trying to find the supremum of the following set $$\{xy−x^2/2\mid x\in [-1,1]\}$$, where $$y$$ is a real number. I am not sure if this is correct, but I managed to find that $$\sup \{ xy-x^2/2\mid x\in [-1,1]\}=\begin{cases} y^2/2 & \text{ if } y\in [-1,1] \ y-1/2 & \text{ if } y>1 \ -y-1/2 & \text{ if } y<-1 \end{cases}$$