Question

If f (x)= (x-1)(1-3) on [1,3] , then the suitable C in [1,3] for the Rolle's theorem such that fcas=0 is
A] √2
B]√3
C]2
D]4​

Answer 1

$\underline{\textbf{Given:}}$

$\textsf{f(x)=(x-1)(x-3) on [1,3]}$

$\underline{\textbf{To find:}}$

$\textsf{The value of C according to Rolle's theorem}$

$\underline{\textbf{Solution:}}$

$\mathsf{f(x)=(x-1)(x-3)=x^2-4x+3}$

$\mathsf{f'(x)=2x-4}$

$\textsf{Since f(x) is a polynomial,}$

$\textsf{(i) f(x) is continuous on [1,3] }$

$\textsf{(ii) f(x) is differentiable in (1,3) }$

$\therefore\textsf{Conditions of Rolle's theorem are satisfied}$

$\textsf{By Rolle's theorem there exists a}\;\mathsf{c\,\in\,(1,3)}$

$\textsf{such that f'(c)=0}$

$\implies\mathsf{2c-4=0}$

$\implies\mathsf{2c=4}$

$\implies\mathsf{c=2\;\;and\;\;2\,\in\,(1,3)}$

$\underline{\textbf{Answer:}}$

$\textsf{Option (C) is correct}$