jmc

We factor the quadratic, getting $(b-5)(2-b)\ge 0$. The expression is equal to $0$ when $b=5\text{ or }2$. When $b\le 2$ or $b\ge 5$, the quadratic is negative. When $2\le b\le 5$, the quadratic is non-negative. Therefore, the greatest value of $b$ for which $(b-5)(2-b)\ge 0$ is $b=\boxed{5}$.