We establish the mean-field bounds $γ\ge 1$, $δ\ge 2$ and $\triangle \ge 2$ on the critical exponents of the Poisson-Boolean continuum percolation model under a moment condition on the radii; these were previously known only in the special case of fixed radii (in the case of $γ$), or not at all (in the case of $δ$ and $\triangle$). We deduce these as consequences of the mean-field bound $β\le 1$, recently established by Duminil-Copin, Raoufi and Tassion under the same moment condition, using a relative entropy method introduced by the authors in previous work.