$A$ value of $c$ for which the conclusion of the Mean Value Theorem holds for the function $f(x) = \log_{e}x$ on the interval $[1, 3]$ is

If $f:[a, b] \rightarrow [c, d]$ is a continuous and strictly increasing function, then $\frac{d-c}{b-a}$ is

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?

If $f(x)$ is differentiable on the interval $[2, 5]$ such that $f(2) = 1/5$ and $f(5) = 1/2$,then there exists a number $c$ such that $2 < c < 5$ and $f'(c) = \dots$

If $f(x) = \cos x$ for $0 \le x \le \frac{\pi}{2}$,then the real number $c$ of the Mean Value Theorem is:

If $x = \alpha$ is a positive root of the equation $a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x = 0$,then what can be said about the positive root of the equation $na_nx^{n-1} + (n-1)a_{n-1}x^{n-2} + \dots + a_1 = 0$?