$f:[1,3] \rightarrow R$ is a function defined as $f(x)=x^3+a x^2+b x$. If $f(1)-f(3)=0$ and $f^{\prime}\left(\frac{2 \sqrt{3}+1}{\sqrt{3}}\right)=0$,then $a-b$ is equal to

If the function $f(x)=\sqrt{x^2-4}$ satisfies the Lagrange's mean value theorem on $[2, 4]$,then the value of $C$ is

Let $a > 0$ and $f$ be continuous in $[-a, a]$. Suppose that $f'(x)$ exists and $f'(x) \le 1$ for all $x \in (-a, a)$. If $f(a) = a$ and $f(-a) = -a$,then $f(0)$ is:

The number of values of $C$ that satisfy the conclusion of Rolle's theorem for the function $f(x) = \sin(2 \pi x)$ on the interval $x \in [-1, 1]$ is:

$A$ value of $c$ for which the conclusion of the Mean Value Theorem holds for the function $f(x) = \log_e x$ on the interval $[1, 3]$ is

Let $S$ be the set of all functions $f:[0,1] \rightarrow \mathbb{R}$ which are continuous on $[0,1]$ and differentiable on $(0,1)$. Then for every $f \in S$,there exists a $c \in (0,1)$,depending on $f$,such that: