Question

Let $f_{1}:(0, \infty) \rightarrow R$ and $f_{2}:(0, \infty) \rightarrow R$ be defined by
$f_{1}(x)=\int\limits_{0}^{x} \displaystyle\prod_{j=1}^{21}(t-j)^{j} d t,$ x$>$0
and $f _{2}( x )=98( x -1)^{50}-600( x -1)^{49}+2450, x >0$,
where, for any positive integer $n$ and real numbers $a _{1}, a _{2}$, $\ldots , a_{n}, \displaystyle\prod_{i=1}^{n} a_{i}$ denotes the product of $a_{1}, a_{2}, \ldots , a_{n}$. Let $m_{i}$ and $n_{i}$, respectively, denote the number of points of local minima and the number of points of local maxima of function $f _{ i }, i =1,2$, in the interval $(0$, $\infty$ )