Prove that critical damping converges fastest to equilibrium postion. I have just read this PDF from ocw.mit.edu. 
How to prove that critical damping gives the fastest return to the equilibrium position? 
I thought of proving that it converges faster than 1) the overdamped case & 2) the underdamped case. The overdamped case seems to be conceptually (with the two roots) clear but not how to prove it under all initial conditions. I don't know how to start with the underdamped case since the exponent is the same as in the critical case.
How to prove that the assumption is true under all initial conditions? I don't really know how to deal with the $(c_1+c_2t)$, especially the $t$, in the critical damping solution $$x=(c_1+c_2t)e^{-bt/2m}$$
I would appreciate any help.
 A: You have simply the General solution of the differential equation; this is given in dependence of two constants $c_1,c_2$ and holds for all initial condition. For example the solution of the critical damped System is:
$(c_1+c_2t)e^{-bt/(2m)}$.
This holds for every initial conditions (requiring that the System is damped critically); any initial condition can be computed with this equation (one has only to adjust $c_1,c_2$). The $t$ can be understood as the time elapsed after the System Begins to oscillate. From the duration of the oscillation $T$ you can compute the ratios $\frac{x(t+T)}{x(t)}$ and look how small These ratios are to analyze how fast the System tends to Equilibrium.
The factor $(c_1+c_2t)$ arises as the most General solution of the critically damped oscillation equation.
A: In the overdampened case, if we set $\lambda = \frac{b}{2m}$, $\omega_0 = \sqrt\frac{k}{m}$, and $a = \sqrt{\lambda^2 - \omega_0^2}$, then we have
$\lim\limits_{t \to \infty} \frac{(c_1 + c_2t)e^{-\lambda t}}{c_3e^{-(\lambda + a)t} + c_3e^{-(\lambda - a)t}}$ 
$= \lim\limits_{t \to \infty} \frac{(c_1 + c_2t)}{c_3e^{-at} + c_3e^{+at}}$
$\stackrel{\text{L'H}}{=} \lim\limits_{t \to \infty} \frac{c_2}{-ac_3e^{-at} + ac_3e^{+at}}$
$= 0$
so the critically-dampened case goes to the equilibrium position faster than any of the overdampened cases.
In the underdamped case, the mass actually reaches the equilibrium position in a finite amount of time, whereas in the overdampened case, the mass never quite reaches the equilibrium position in finite time.
