A particle moves along the $x$-axis with position at time $t$ given by $x(t) = e^{-t}\sin t$ for $0 \leq t \leq 2\pi$.
(a) Find the time $t$ at which the particle is farthest to the left. Justify your answer.
(b) Find the value of the constant $A$ for which $x(t)$ satisfies the equation $Ax^{\prime\prime}(t) + x^{\prime}(t) + x(t) = 0$ for $0 < t < 2\pi$.