#### Need solution for RD Sharma math class class 12 chapter Mean value theorem exercise 14.2 question 2 sub question  (ii)

$c=2 \in(1,3)$ , Hence, Rolle’s Theorem is verified.

Hint:

$f(x)$is continuous for all x and hence continuous in [1,3]

Given:

$f(x)=x^{2}-4 x+3$ on [1,3]

Explanation:

We have

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

1. Being polynomial f(x) is continuous for all x and hence continuous in [1,3]
2. $f^{\prime}(x)=2 x-4$ , which exists in (1,3)

$\therefore f (x )$is derivable in (1,3)

3.

$\\f(1)=(1)^{2}-4(1)+3\\\\ =1-4+3=-3+3=0\\\\ f(3)=(3)^{2}-4(3)+3\)\\\\\ =9-12+3=-3+3=0\\\\ \therefore f(1)=f(3)$

Thus all the conditions of Rolle’s Theorem are satisfied.

There exists at least one$c \in(1,3)$such that$f'(c) = 0$

$\\ \Rightarrow 2c - 4 = 0 \\\\ \Rightarrow c = 2 \\\\ \Rightarrow c = 2 \in (1,3)$

Hence, Rolle’s Theorem is verified.