g(t)=12e-t²,
g'(t)=-24te-t² is zero at an extremum, that is, when t=0, the point (0,12).
g''(t)=-24e-t²+48t2e-t². g''(0)=-24. This is negative concavity so there is a maximum at (0,12).
As t→±∞, g(t)→0. The t-axis is the asymptote. (0,12) is also an absolute maximum. There is no absolute minimum because g(t) cannot be equal to zero.