For which interval does the function $f(x) = \frac{x^2 - 3x}{x - 1}$ satisfy all the conditions of Rolle's theorem?

Examine if Rolle's Theorem is applicable to the function $f(x) = [x]$ for $x \in [5, 9]$. Can you say something about the converse of Rolle's Theorem from this example?

If $c$ is a point at which Rolle's theorem holds for the function $f(x) = \log_{e}\left(\frac{x^{2}+\alpha}{7x}\right)$ in the interval $[3, 4]$,where $\alpha \in R$,then $f''(c)$ is equal to

If $c = \frac{1}{2}$ and $f(x) = 2x - x^2$,then the interval $(a, b)$ of $x$ in which the Lagrange's Mean Value Theorem $(LMVT)$ is applicable for the function $f(x)$ is:

Verify Rolle's Theorem for the function $f(x)=x^{2}+2x-8, x \in[-4,2]$

Find the value of $p$ and $q$ if the function $f(t) = t^3 - 6t^2 + pt + q$ defined on $[1, 3]$ satisfies Rolle's theorem for $c = \frac{2\sqrt{3} + 1}{\sqrt{3}}$.