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

If the function $f(x) = 2x^2 + 3x + 5$ satisfies the Lagrange's Mean Value Theorem $(LMVT)$ at $x = 3$ on the closed interval $[1, a]$,then the value of $a$ is equal to:

Let $f:[a, b] \rightarrow R$ be such that $f$ is differentiable in $(a, b)$,continuous at $x=a$ and $x=b$,and $f(a)=0=f(b)$. Then:

If the function $f(x) = x^3 - 6x^2 + ax + b$ defined on $[1, 3]$ satisfies Rolle's theorem for $c = \frac{2\sqrt{3} + 1}{\sqrt{3}}$,then:

Let $f:[1,3] \rightarrow R$ be continuous and differentiable in $(1,3)$ such that $f^{\prime}(x)=[f(x)]^2+4$ for all $x \in (1,3)$. Then:

Let $f(x)$ be a non-constant twice differentiable function defined on $(-\infty, \infty)$ such that $f(x)=f(1-x)$ and $f^{\prime}\left(\frac{1}{4}\right)=0$. Then
$(A)$ $f^{\prime \prime}(x)$ vanishes at least twice on $[0,1]$
$(B)$ $f^{\prime}\left(\frac{1}{2}\right)=0$
$(C)$ $\int_{-1 / 2}^{1 / 2} f\left(x+\frac{1}{2}\right) \sin x d x=0$
$(D)$ $\int_0^{1 / 2} f(t) e^{\sin \pi t} d t=\int_{1 / 2}^1 f(1-t) e^{\sin \pi t} d t$