Q. If $f(x) = 1 + nx + \frac{n(n - 1)}{2}x^{2} + \frac{n(n - 1)(n - 2)}{6}x^{3} + .\ldots + x^{n},then\ f^{''}(1)$ is equal to
A) $n(n - 1)2^{n - 1}$
B) $(n - 1)2^{n - 1}$
C) $n(n - 1)2^{n - 2}$
D) $n(n - 1)2^{n}$
Explanation
(c)
Given, $f(x) = 1 + nx + \frac{n(n - 1)}{2!\ }x^{2}$
$+ \frac{n(n - 1)(n - 2)}{3!}x^{3} + ... + x^{n}$
$\Rightarrow \$ $f(x) = (1 + x)^{n}$
$\Rightarrow f^{'}(x) = n(1 + x)^{n - 1}$
$\Rightarrow f^{''}(x) = n(n - 1)(1 + x)^{n - 2}$
$\Rightarrow f^{''}(1) = n(n - 1)2^{n - 2}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.