Limit property of second derivative of bounded function.
Solution 1:
As it was suggested in a previous comment, assume by contradiction that $t^2f(t)\geq\varepsilon$ for all $t\geq M$. Divide by $t^2$ the inequality, integrate between $y$ and $x$ (both bigger than $M$) and multiply everything by $y$ so that you obtain
$$yf^\prime(y)\geq y\biggl(\frac{xf^\prime(x)+\varepsilon}{x}\biggr)-\varepsilon.$$
Choose $x$ so big (and $y$ bigger) that $|f^\prime(x)x|\leq\varepsilon/2$ and in this way you get a contradiction.