jmc

Since $|f(x)|\le 5$ for all $x$ and $|g(x)|\le 2$ for all $x$, $|f(x)g(x)|\le 10$ for all $x$. It follows that $f(x)g(x)\le 10$ for all $x$, so $b$ is at most 10.

Furthermore, if $f$ is any function such that the range of $f$ is $[-5,3]$ and $f(0)=-5$, and $g$ is any function such the range of $g$ is $[-2,1]$ and $g(0)=-2$, then $f(0)g(0)=(-5)\cdot (-2)=10$. Therefore, the largest possible value of $b$ is $\boxed{10}$.