Is the product of a positive variable and a convex function also convex?

Question

I have a convex function $f:R_{\geq0}^N\rightarrow R$ and its first order derivatives are all positive. Now I find it is difficult to prove that the convexity of the function $g(x,y)=xf(y)$, where $x\in R_{\geq0}$ and $y\in R^N_{\geq0}$. Anybody knows how to do it?