I have the following function $$f(t) = \frac{1}{\gamma}e^{-t/\gamma}\left(\frac{t}{\gamma} - \frac{t^2}{2\gamma^2} \right) $$
We have the property that $\int_0^\infty f(t) dt = 0$. Also, the function has a max at $t_1=\gamma(2-\sqrt{2})$ and a min at $t_2=\gamma(2+\sqrt{2})$.
What I want to show is that when $\gamma\to 0$, the maximum and the minimum becomes two delta functions. Something like, $f(t) \to \delta(t-t_1)- \delta(t-t_2)$ up to some proportionality.
Could someone help with this?