= Solution
{c}
For the one-loop <renormalization-group beta function>, write $t=\log\mu$. Separating variables gives
$$
\int_{\alpha_i}^{\alpha_f}\frac{d\alpha}{\alpha^2}=b_0\int_{\log\mu_i}^{\log\mu_f}dt,
\qquad
\boxed{\alpha(\mu_f)=\frac{\alpha(\mu_i)}{1-b_0\alpha(\mu_i)\log(\mu_f/\mu_i)}.}
$$
This expression resums the leading logarithms of the one-loop <running coupling>, while omitting higher-order terms in the <renormalization-group beta function>. Assume a positive initial coupling.
For $b_0<0$, increasing the <renormalization scale> decreases the <running coupling>; the zero-coupling <renormalization-group fixed point> is attractive in the ultraviolet. This is <asymptotic freedom>. Toward lower scales the one-loop expression has a formal pole at $\mu_i\exp[1/(b_0\alpha_i)]<\mu_i$. The strong-coupling neighborhood of that pole lies outside perturbative control. <QCD> provides the main example; the minimal <Standard Model> $SU(2)$ coupling is also <asymptotically free> at one loop.
For $b_0>0$, the <running coupling> grows with the <renormalization scale> and formally diverges at the ultraviolet <Landau pole> $\mu_i\exp[1/(b_0\alpha_i)]>\mu_i$. Toward the infrared it decreases to zero within this approximation. The <Standard Model> $U(1)$ <weak hypercharge> coupling has this sign, as does <QED> in a fixed charged-particle regime. A perturbative <Landau pole> marks failure of the approximation, rather than proving that the exact theory has that divergence.
For $b_0=0$, the <running coupling> is constant at this order; higher-loop terms decide whether it runs. Vanishing of the leading coefficient alone does not establish exact scale invariance.
For a concrete common convention, define $\alpha_i=g_i^2/(4\pi)$ with $g_Y$ in its usual, non-unified <weak hypercharge> normalization. In the minimal <Standard Model> with three generations and one Higgs doublet, above its particle thresholds,
$$
\frac{d\alpha_i}{d\log\mu}=\frac{b_i}{2\pi}\alpha_i^2,\qquad
(b_Y,b_2,b_3)=\left(\frac{41}{6},-\frac{19}{6},-7\right).
$$
Thus the question's $b_0$ is $b_i/(2\pi)$, with the sign regimes just described. Crossing a <mass> threshold changes the active field content and requires the appropriate piecewise <renormalization-group beta function>.
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