= Solution
Let $S$ denote this maximum and put $\tau=\tau(-b)$. <Brownian motion> reaches $-b$ almost surely: the <Brownian reflection principle> gives crossing probability $2\mathbb P(W_t<-b)\to1$. Continuity gives $W_\tau=-b$, with the strict-crossing infimum interpreted as in part (b). The process
$$
M_t=\frac{W_{t\wedge\tau}+b}{b}
$$
is a continuous nonnegative <local martingale> starting at one and tending to zero. Its maximum is $1+S/b$. Apply part (b) at $a=1+x/b$, for $x>0$:
$$
\mathbb P(S>x)=\frac b{b+x},\qquad\mathbb P(S\leq x)=\frac x{b+x}.
$$
Differentiating gives the <maximum before a lower Brownian barrier> density
$$
\boxed{f_S(x)=\frac b{(b+x)^2}\mathbf1_{\{x>0\}}.}
$$
The tail tends to one as $x\downarrow0$, so there is no atom at zero. The density integrates to one.
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