Verifying the convexity of some function Convex function: We will say that $f:X\rightarrow R$ is convex function if for every $\lambda\in [0,1]$ and for every $x,y\in X$ ($X$ is convex space) $f(\lambda x+(1-\lambda)y)\leq\lambda f(x)+(1-\lambda)f(y)$.
We have $F:R_{++}^2\rightarrow R$ by $F(x_1,x_2)=-x_1x_2-\alpha$ in which $R_{++}^2=[0,+\infty)\times [0,+\infty)$ and $\alpha\in R$ is arbitrary. Is $F$ function convex?

This is most easily answered by checking whether the Hessian is positive semidefinite.
The Hessian is defined as $\nabla^2f(x)$ where $[\nabla^2f(x)]_{ij} = \dfrac{\partial^2 f}{\partial x_i \partial x_j}$, and in this case is easy to compute.
