Let f be a real valued function defined on $\mathrm{ℝ}\text{ such that }{\mathrm{f}}^{\mathrm{\prime }}\left(\mathrm{x}\right)=\prod _{\mathrm{r}=1}^{2020} \left(\mathrm{x}-{\mathrm{r}}^{\mathrm{r}}\right)$ and g is a function defined on $\mathrm{ℝ}\text{ such that }{\int }_0^{\mathrm{x}} \left({\mathrm{e}}^{\mathrm{f}\left(\mathrm{t}\right)}-\mathrm{g}(\mathrm{x}-\mathrm{t})\mathrm{dt}=2019\mathrm{x}\right)+2020, \mathrm{then}$

$$\begin{gathered} {\mathrm{f}}^{\mathrm{\prime }}\left(\mathrm{x}\right)=(\mathrm{x}+1)(\mathrm{x}-2{)}^2(\mathrm{x}-3{)}^3\dots .(\mathrm{x}-2020{)}^{2020} \\ {\int }_0^{\mathrm{x}} {\mathrm{e}}^{\mathrm{f}\left(\mathrm{t}\right)}\mathrm{dt}-{\int }_0^{\mathrm{x}} \mathrm{g}(\mathrm{x}-\mathrm{t})\mathrm{dt}=2019\mathrm{x}+2020 \\ {\int }_0^{\mathrm{x}} {\mathrm{e}}^{\mathrm{f}\left(\mathrm{t}\right)}\mathrm{dt}-{\int }_0^{\mathrm{x}} \mathrm{g}(\mathrm{t})\mathrm{dt}=2019\mathrm{x}+2020\text{ diff. both sides } \\ {\mathrm{e}}^{\mathrm{f}\left(\mathrm{x}\right)}=\mathrm{g}(\mathrm{x})+2019 \end{gathered}$$
diff again $\Rightarrow {\mathrm{e}}^{\mathrm{f}\left(\mathrm{x}\right)}\cdot {\mathrm{f}}^{\mathrm{\prime }}\left(\mathrm{x}\right)={\mathrm{g}}^{\mathrm{\prime }}\left(\mathrm{x}\right)\Rightarrow$nature of monotoncity of g(x) is same as that of f(x)
 

$$\begin{gathered} \mathrm{local}\max \mathrm{at} \mathrm{x}\in \{1,5,9\dots .2017\}505 \mathrm{points} \\ \mathrm{local}\min \mathrm{at} \mathrm{x}\in \{3,7,11,2019\}505 \mathrm{points} \end{gathered}$$