Concavity of function with 2 variables

Question

I was thinking if I have a function of 2 variables $F(x,y)$ and if the 2nd order derivative with respect to both variables are negative, i.e., $F_{xx}<0$ and $F_{yy}<0$. Then, geometrically, this means for any fixed $y_0$, the function $F(x,y_0)$ is concave in $x$ and for any fixed $x_0$, the function $F(x_0,y)$ is concave in $y$ right? Then I feel that geometrically, this function $F(x,y)$ should have global maximum point right? (may be unqiue?) And this function should not have saddle point right? (I cannot picture that situation). So can I safely claim any stationary point $(x_0,y_0)$ such that $F_x(x_0,y_0)=0$ and $F_y(x_0,y_0)=0$ must be the global maximum point? What is the role of the $F_{xy}$, i.e., the cross partial derivative plays here and why we need to check Hessian in this case? How to think about Hessian geometrically?