Question:
Let $f: \mathrm{R} \rightarrow \mathrm{R}$ be a differentiable function satisfying $f^{\prime}(3)+f^{\prime}(2)=0$.
Correct Option: , 4
Solution:
$\lim _{x \rightarrow 0}\left(\frac{1+f(3+x)-f(3)}{1+f(2-x)-f(2)}\right)^{\frac{1}{x}}\left(1^{\infty}\right.$ form $)$
$\Rightarrow \mathrm{e}^{\lim _{x \rightarrow 0} \frac{f(3+x)-f(2-x)-f(3)+f(2)}{x(1+f(2-x)-f(2))}}$
using L'Hopital
$\Rightarrow e^{\lim _{x \rightarrow 0} \frac{f(3+x)+f(2-x)}{-x f^{\prime}(2-x)+(1+f(2-x)-f(2))}}$
$\Rightarrow \mathrm{e}^{\frac{f^{\prime(3)+f^{\prime}(2)}}{1}}=1$