  We investigate the extreme value statistics of a one-dimensional Brownian
motion (with the diffusion constant $D$) during a time interval $\left[0, t
\right]$ in the presence of a reflective boundary at the origin, starting from
a positive position $x_0$. By deriving the survival probability of the Brownian
particle without hitting an absorbing boundary at $x=M$, we obtain the
distribution $P(M|x_0,t)$ of the maximum displacement $M$ and its expectation
$\langle M \rangle$. In the short-time limit, i.e., $t \ll t_d$ where
$t_d=x_0^2/D$ is the diffusion time from the starting position $x_0$ to the
reflective boundary at the origin, the particle behaves like a free Brownian
motion without any boundaries. In the long-time limit, $t \gg t_d$, $\langle M
\rangle$ grows with $t$ as $\langle M \rangle \sim \sqrt{t}$, which is similar
to the free Brownian motion, but the prefactor is $\pi/2$ times of the free
Brownian motion, embodying the effect of the reflective boundary. By solving
the propagator and using a path decomposition technique, we obtain the joint
distribution $P(M,t_m|x_0,t)$ of $M$ and the time $t_m$ at which this maximum
is achieved, from which the marginal distribution $P(t_m|x_0,t)$ is also
obtained. For $t \ll t_d$, $P(t_m|x_0,t)$ looks like a U-shaped attributed to
the arcsine law of free Brownian motion. For $t$ equal to or larger than order
of magnitude of $t_m$, $P(t_m|x_0,t)$ deviates from the U-shaped distribution
and becomes asymmetric with respect to $t/2$. Moreover, we compute the
expectation $\langle t_m \rangle$ of $t_m$, and find that $\langle t_m
\rangle/t$ is an increasing function of $t$. In two limiting cases, $\langle
t_m \rangle/t \to 1/2$ for $t \ll t_d$ and $\langle t_m \rangle/t \to (1+2G)/4
\approx 0.708$ for $t \gg t_d$, where $G\approx0.916$ is the Catalan's
constant. All the theoretical results are validated by numerical simulations.
