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:

If $f(x)=x^3+p x^2+q x$ is defined on $[0,2]$ such that $f(0)=f(2)$ and $f^{\prime}\left(1+\frac{1}{\sqrt{3}}\right)=0$,then $p^2+q^2=$

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:

If $f(x) = \sqrt{x}$ and $g(x) = \frac{1}{\sqrt{x}}$ for $x \in [3, 12]$,then the value of $c \in (3, 12)$ for which $\frac{f^{\prime}(c)}{g^{\prime}(c)} = \frac{f(12) - f(3)}{g(12) - g(3)}$ holds,is

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

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 $[0,1]$
$(B)$ $f^{\prime}\left(\frac{1}{2}\right)=0$
$(C)$ $\int_{-1 / 2}^{1 / 2} f\left(x+\frac{1}{2}\right) \sin x d x=0$
$(D)$ $\int_0^{1 / 2} f(t) e^{\sin \pi t} d t=\int_{1 / 2}^1 f(1-t) e^{\sin \pi t} d t$