Let $f(x) = \begin{cases} \frac{x^p}{(\sin x)^q} & \text{if } 0 < x \leq \frac{\pi}{2} \\ 0 & \text{if } x = 0 \end{cases}$ where $p, q \in \mathbb{R}$. Then,Lagrange's Mean Value Theorem is applicable to $f(x)$ in the closed interval $[0, \frac{\pi}{2}]$ if:

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?

If the function $f(x) = ax^3 + bx^2 + 26x - 24$ satisfies the conditions of Rolle's theorem in $[2, 4]$ and $f^{\prime}\left(3 + \frac{1}{\sqrt{3}}\right) = 0$,then the value of $ab$ is equal to

Let $f(x) = e^x \cos x + 1$. Which of the following statements is always true?

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

Let $f^{\prime}(0)=-3$ and $f^{\prime}(x) \leq 5$ for all real values of $x$. The $f(2)$ can have possible maximum value as