The value $c$ of Lagrange's mean value theorem for $f(x)=e^{x}+24$ in $[0,1]$ is

The value of $c$ in $(0, 2)$ satisfying the Mean Value Theorem for the function $f(x) = x(x - 1)^2, x \in [0, 2]$ is equal to

Suppose $f''(x)$ exists for all real $x$. If $f(2) = 2$,$f(3) = 5$ and $f(4) = 10$,then which one among the following statements is definitely true?

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?

The number of admissible values of $C$ obtained when the Lagrange's mean value theorem is applied for the function $f(x)=x$ on $[2,5]$ is

Let $f$ be any continuously differentiable function on $[a, b]$ and twice differentiable on $(a, b)$ such that $f(a)=f^{\prime}(a)=0$ and $f(b)=0$. Then: