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 $y = f(x)$ and $y = g(x)$ be two differentiable functions in $[0, 2]$ such that $f(0) = 3$,$f(2) = 5$,$g(0) = 1$,and $g(2) = 2$. If there exists at least one $c \in (0, 2)$ such that $f'(c) = k g'(c)$,then $k$ must be:

If the function $f(x)=x^3+ax^2+bx+40$ satisfies the conditions of Rolle's theorem on the interval $[-5,4]$ and $-5,4$ are two roots of the equation $f(x)=0$,then one of the values of $c$ as stated in that theorem is

Let $f(x)$ satisfy all the conditions of the Mean Value Theorem in $[0, 2]$. If $f(0) = 0$ and $|f'(x)| \le \frac{1}{2}$ for all $x$ in $[0, 2]$,then:

Let $f$ and $g$ be differentiable on the interval $I$ and let $a, b \in I, a < b$. Then,

Let $f(x)$ be differentiable on $[1, 6]$ and $f(1) = -2$. If $f(x)$ has only one root in $(1, 6)$,then there exists $c \in (1, 6)$ such that: