What is the minimum value of the function $f(x)={\displaystyle \underset{0}{\overset{x}{}}}(t-1)(t-2)dt$ where $x>0?$

$($a$)1$

$($b$)\frac{1}{2}$

$($c$)\frac{2}{3}$

$($d$)\frac{5}{2}$