JEE Advanced 2021 — Paper 2 — Question 45

Let $f_{1}:(0, \infty) \rightarrow R$ and $f_{2}:(0, \infty) \rightarrow R$ be defined by $f_{1}(x)=\int_{0}^{x} \prod_{j=1}^{21}(t-j)^{j} d t, x>0$ and $f_{2}(x)=98(x-1)^{50}-600(x-1)^{49}+2450, x>0$, where, for any positive integer $n$ and real numbers $a_{1}, a_{2}$, $\ldots \ldots, a_{n}, \prod_{i=1}^{n} a_{i}$ 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 , $\infty)$.

The value of $2 m_{1}+3 n_{1}+m_{1} n_{1}$ is $\qquad$