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

If the function $f(x)=x^3+b x^2+c x-6$ satisfies all the conditions of Rolle's theorem in $[1,3]$ and $f^{\prime}\left(\frac{2 \sqrt{3}+1}{\sqrt{3}}\right)=0$,then $b c=$

If $a, b, c \in \mathbb{R}$ and satisfy $3a + 5b + 15c = 0$,then the equation $ax^4 + bx^2 + c = 0$ has:

Examine the applicability of the Mean Value Theorem for the following functions:
$(i)$ $f(x) = [x]$ for $x \in [5, 9]$
$(ii)$ $f(x) = [x]$ for $x \in [-2, 2]$
$(iii)$ $f(x) = x^{2} - 1$ for $x \in [1, 2]$

In which of the following functions is Rolle's theorem applicable?

The value of $c$ for the function $f(x) = \log x$ on $[1, e]$ if Lagrange's Mean Value Theorem $(LMVT)$ is applied,is