$f:[2,10] \rightarrow R$ is defined as $f(x) = \begin{cases} \frac{1}{2}(x-6)^2-3, & x \leq 4 \\ x-5, & x > 4 \end{cases}$. Which of the following is true?

For what value of $c$ does the conclusion of the Mean Value Theorem hold for the function $f(x) = \log_e x$ on the interval $[1, 3]$?

Which of the following functions can satisfy Rolle's theorem on the given interval?

Let $f(x)$ be continuous on $[0,6]$ and differentiable on $(0,6)$. Let $f(0)=12$ and $f(6)=-4$. If $g(x)=\frac{f(x)}{x+1}$, then for some Lagrange's constant $c \in(0,6)$, $g^{\prime}(c)=$

If a polynomial equation $a_nx^n + a_{n-1}x^{n-1} + \dots + a_2x^2 + a_1x + a_0 = 0$,where $n$ is a positive integer,has two distinct roots $\alpha$ and $\beta$,then how many roots does the equation $na_nx^{n-1} + (n - 1)a_{n-1}x^{n-2} + \dots + a_1 = 0$ have in the interval $(\alpha, \beta)$?

Consider the following statements:
Statement $I$: If $a_0+\frac{a_1}{2}+\frac{a_2}{3}+\ldots+\frac{a_n}{n+1}=0$, where $a_0, a_1, \ldots, a_n$ are real numbers, then the polynomial $P(x) = a_0+a_1 x+a_2 x^2+\ldots+a_n x^n$ has a zero in the interval $(0,1)$.
Statement $II$: If $f:[a, b] \rightarrow R$ is continuous on $[a, b]$ and $f$ is differentiable in $(a, b)$, where $a>0$ and if $\frac{f(a)}{a}=\frac{f(b)}{b}$, then there exists $c \in(a, b)$ such that $c f^{\prime}(c)=f(c)$.
Which one of the following options is true?