Question bank/JEE Advanced/Mathematics

Let $f_1:(0, \infty) \rightarrow \mathbb{R}$ and $f_2:(0, \infty) \rightarrow \mathbb{R}$ 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 $\mathrm{n}$ and real numbers $\mathrm{a}_1, \mathrm{a}_2, \ldots, \mathrm{a}_{\mathrm{n}}, \prod_{i=1}^{\mathrm{n}} \mathrm{a}_i$ denotes the product of $\mathrm{a}_1, \mathrm{a}_2, \ldots, \mathrm{a}_{\mathrm{n}}$. Let $\mathrm{m}_i$ and $\mathrm{n}_i$, 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 $(0, \infty)$. What is the value of $2 m_{1}+3 n_{1}+m_{1} n_{1}$?

Source: JEEBench (IIT Delhi) / JEE Adv 2021 Paper 2