In the Mean Value Theorem,$f(b) - f(a) = (b - a)f'(c)$. If $a = 4$,$b = 9$ and $f(x) = \sqrt{x}$,then the value of $c$ is:

Consider all functions given in List-$I$ in the interval $[1,3]$. List-$II$ has the values of '$c$' obtained by applying Lagrange's Mean Value Theorem $(LMVT)$ on the functions of List-$I$. Match the functions and values of '$c$'.

List-$I$	List-$II$
$A. |x-1|$	$I. 2 \log (e^3+e^2)$
$B. \log x$	$II. 2$
$C. x^2+x+1$	$III. \log_3 e^2$
$D. e^x$	$IV. \sqrt{2}$
	$V. \log \left(\frac{e^3-e}{2}\right)$

If the function $f(x)=x^3+ax^2+bx+40$ satisfies the conditions of Rolle's theorem on the interval $[-5,4]$ and $-5,4$ are two roots of the equation $f(x)=0$,then one of the values of $c$ as stated in that theorem is

The value of $c$ in the Lagrange's mean value theorem for $f(x)=\sqrt{x-2}$ in the interval $[2,6]$ is

Let $f(x)$ be continuous on $[0,4]$,differentiable on $(0,4)$,$f(0)=4$ and $f(4)=-2$. If $g(x)=\frac{f(x)}{x+2}$,then the value of $g^{\prime}(c)$ for some Lagrange's constant $c \in (0,4)$ is

The value of $C$ in the Mean Value Theorem for the function $f(x) = x^{2}$ in the interval $[2, 4]$ is: