JEE Main 2025 — 22 January, Morning Shift — Question 16

$f(x+y)=f(x) f^{\prime}(y)+f^{\prime}(x) f(x)$

Put $=x=y=0$ $f(0)=f(0) f^{\prime}(0)+f^{\prime}(0) f(0)$

$\mathrm{f}^{\prime}(0)=\frac{1}{2}$

Put $y=0$ $f(x)=f(x) f^{\prime}(0)+f^{\prime}(x) f(0)$

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

$f^{\prime}(x)=\frac{f(x)}{2}$

$\frac{d y}{d x}=\frac{y}{2} \Rightarrow \int \frac{d y}{y}=\int \frac{d x}{2}$

$\Rightarrow \ell$ ny $=\frac{x}{2}+c$

$\because \mathrm{f}(0)=1 \Rightarrow \mathrm{C}=0$

$\ell$ ny $=\frac{\pi}{2} \Rightarrow f(x)=e^{x / 2}$

$\ell n f(n)=\frac{n}{2}$

$\sum_{\mathrm{n}=1}^{100} \ell \mathrm{f}(\mathrm{n})=\frac{1}{2} \sum_{\mathrm{n}=1}^{100} \mathrm{n}=\frac{5050}{2}$

$=2525$