The function $f(x) = x^3 - 6x^2 + ax + b$ satisfies the conditions of Rolle's theorem in $[1, 3]$. Then the values of $a$ and $b$ are respectively

If $f(x) = x^2 - 2x + 4$ and $\frac{f(5) - f(1)}{5 - 1} = f'(c)$,then the value of $c$ will be

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

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a twice continuously differentiable function such that $f(0)=f(1)=f^{\prime}(0)=0$. Then:

The function $f(x) = (x - 3)^2$ satisfies all the conditions of the Mean Value Theorem in $[3, 4]$. $A$ point on $y = (x - 3)^2$,where the tangent is parallel to the chord joining $(3, 0)$ and $(4, 1)$,is

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