Convex function? I have a positive function $f(x,y)$, where $x\in{\mathbb R}^n$ and $y\in{\mathbb R}$. I know that for $y$ fixed, $g(x)=f(x,y)$ is convex, and that for $x$ fixed, $h(y)=f(x,y)$ has positive second derivative. If this enough to show that $f(x,y)$ is convex?

Not quite. Notice that if $f$ is twice continuously-differentiable, it must also be that $f_{xx}$ is non-negative, and a condition that a $2-$D function is convex at a point is that it's Hessian $D^2f(a)=\left(\begin{array}{cc}f_{xx} & f_{xy}\\f_{xy} & f_{yy}\end{array}\right)$ is positive semi-definite. Since we know $f_{xx},f_{yy}\geq0$, it must be then that $f_{xx}f_{yy}-f_{xy}^2\geq0$ as well, so we shouldn't expect it to hold in general. In fact, an easy smooth counterexample is $f(x,y)=x^2y^2$.
