We investigate the first-passage properties and extreme-value statistics of an overdamped Brownian particle confined by an external linear potential V(x)=μ∣x−x0∣, where μ>0 is the strength of the potential and x0>0
Nearby in the stack
is the position of the lowest point of the potential, which coincides with the starting position of the particle. The Brownian motion terminates whenever the particle passes through the origin at a random time
tf
. Our study reveals that the mean first-passage time
⟨tf⟩
exhibits a nonmonotonic behavior with respect to
μ
, with a unique minimum occurring at an optimal value of
μ≃1.24468D/x0
, where
D
is the diffusion constant of the Brownian particle. Moreover, we examine the distribution
P(M∣x0)
of the maximum displacement
M
during the first-passage process, as well as the statistics of the time
tm
at which
M
is reached. Intriguingly, there exists another optimal
μ≃1.24011D/x0
that minimizes the mean time
⟨tm⟩
. All our analytical findings are corroborated through numerical simulations.