Proving $t^2 e^{t} - 2 t e^{t} + 2 e^{t} - 2 \geq 0 $
Consider the function $$f(t)=t^2 e^{t} - 2 t e^{t} + 2 e^{t} - 2 $$ for which $f(0)=0$.
What is interesting is that $$f'(t)=t^2 e^{t}$$ is always positive. So ???
Proving $t^2 e^{t} - 2 t e^{t} + 2 e^{t} - 2 \geq 0 $
Consider the function $$f(t)=t^2 e^{t} - 2 t e^{t} + 2 e^{t} - 2 $$ for which $f(0)=0$.
What is interesting is that $$f'(t)=t^2 e^{t}$$ is always positive. So ???