Class 12: Mean Value Theorems

$\displaystyle \textbf{Question 1:}\ \ \text{Verify Rolle's theorem for the function } f(x)=x^2-5x+6 \text{ on the} \\ \text{interval }[2,3]. \text{[CBSE 2002]}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Since a polynomial function is everywhere differentiable and so continuous also.} \\ \text{Therefore, } f(x) \text{ is continuous on }[2,3] \text{ and differentiable on }(2,3).$
$\displaystyle \text{Also, } f(2)=2^2-5\times 2+6=0 \text{ and } f(3)=3^2-5\times 3+6=0$
$\displaystyle \therefore\ f(2)=f(3)$
$\displaystyle \text{Thus, all the conditions of Rolle's theorem are satisfied. Now, we have to show that} \\ \text{there exists some } c\in(2,3), \text{ such that } f'(c)=0.$
$\displaystyle \text{For this we proceed as follows.}$
$\displaystyle \text{We have, } f(x)=x^2-5x+6 \Rightarrow f'(x)=2x-5$
$\displaystyle \therefore\ f'(x)=0 \Rightarrow 2x-5=0 \Rightarrow x=2.5$
$\displaystyle \text{Thus, } c=2.5\in(2,3) \text{ such that } f'(c)=0.\ \text{Hence, Rolle's theorem is verified.}$