JEE Main 2024 — 6 April, Shift 1 — Question 20

f : $(-\infty, \infty)-\{0\} \rightarrow R$

$f^{\prime}(1)=\lim _{a \rightarrow \infty} a^{2} f\left(\frac{1}{a}\right)$

$\lim _{a \rightarrow \infty} \frac{a(a+1)}{2} \tan ^{-1}\left(\frac{1}{a}\right)+a^{2}-2 \ln (a)$

$\lim _{a \rightarrow \infty} a^{2}\left(\frac{\left(1+\frac{1}{a}\right)}{2} \tan ^{-1}\left(\frac{1}{a}\right)+1-\frac{2}{a^{2}} \ln (a)\right)$

$f(x)=\frac{1}{2}(1+x) \tan ^{-1}(x)+1-2 x^{2} \ln (x)$

$\mathrm{f}^{\prime}(\mathrm{x})=\frac{1}{2}\left(\frac{1+\mathrm{x}}{1+\mathrm{x}^{2}}+\tan ^{-1}(\mathrm{x})+4 \mathrm{x} \ln (\mathrm{x})\right)+2 \mathrm{x}$

$\mathrm{f}^{\prime}(1)=\frac{1}{2}\left(1+\frac{\pi}{4}\right)+2$

$\mathrm{f}^{\prime}(1)=\frac{5}{2}+\frac{\pi}{8}$