JEE Main 2024 — 27 January, Shift 1 — Question 20

$\quad f(x)-f(y) \geq \ln x-\ln y+x-y$

$\frac{f(x)-f(y)}{x-y} \geq \frac{\ln x-\ln y}{x-y}+1$

Let $\mathrm{x}>\mathrm{y}$

$\lim _{y \rightarrow x} f^{\prime}\left(x^{-}\right) \geq \frac{1}{x}+1$

Let $\mathrm{x}<\mathrm{y}$

$\lim _{y \rightarrow x} f^{\prime}\left(x^{+}\right) \leq \frac{1}{x}+1$

$f^{\prime}\left(x^{-}\right)=f^{1}\left(x^{+}\right)$

$\mathrm{f}^{\prime}(\mathrm{x})=\frac{1}{\mathrm{x}}+1$

$f^{\prime}\left(\frac{1}{x^{2}}\right)=x^{2}+1$

$\sum_{\mathrm{x}=1}^{20}\left(\mathrm{x}^{2}+1\right)=\sum_{\mathrm{x}-1}^{20} \mathrm{x}^{2}+20$

$=\frac{20 \times 21 \times 41}{6}+20$

$=2890$