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

If the function $f(x) = ax^3 + bx^2 + 26x - 24$ satisfies the conditions of Rolle's theorem in $[2, 4]$ and $f^{\prime}\left(3 + \frac{1}{\sqrt{3}}\right) = 0$,then the value of $ab$ is equal to

If $f(x) = ax^3 + bx^2 + 11x - 6$ for $x \in [1, 3]$ satisfies the conditions of Rolle's theorem and $f'\left( 2 + \frac{1}{\sqrt{3}} \right) = 0$,find $a$ and $b$.

For the function $f(x) = \log(\sin x)$ in the interval $[\frac{\pi}{6}, \frac{5\pi}{6}]$,what is the value of $c$ according to Lagrange's Mean Value Theorem?

Let $f(x)=x^3+2x^2-x$ be a real-valued function. Then, the value of Lagrange's constant $C$ in $(-1,2)$ is

Let $f$ be a function which is differentiable for all real $x$. If $f(2) = -4$ and $f^{\prime}(x) \geq 6$ for all $x \in [2, 4]$, then which of the following is true?