RLC Response (Overdamped, Underdamped, and Critically damped)
Only graph 1 can be critically damped since there is an undershoot, but no subsequent overshoot.

Assuming the capacitor has an initial condition, then the voltage across the three components in parallel is:

\$v=-\frac{1}{C}\large \int \small I_C \: dt \: =L\large\frac{dI_L}{dt}\small  \:=R\left(I_C-I_L\right) \$

Solving the equations simultaneously for \$\small I_L\$ gives:

\$\large \frac{d^2I_L}{dt^2}\small +\large\frac{1}{RC}\frac{dI_L}{dt}\small +\large\frac{1}{LC}\small I_L\$=0

Let \$\small LC=1\$, i.e. natural frequency = 1 rad/sec,  for simplicity, but without losing generality.

Then, for critically damped (equal roots) the solution is of the form:

\$\small I_L=(A+Bt)e^{-t}\$, where \$\small A\$ and \$\small B\$ are constants.

The voltage across the inductor (and R and C) is \$v=\small L\large \frac{dI_L}{dt}\$, giving:

\$v=\small L(B-A-Bt)e^{-t}\$

Which has the characteristic shape of Graph 1