The value of $C$ in the Mean Value Theorem for the function $f(x) = x^{2}$ in the interval $[2, 4]$ is:

Let $f(x)=x^3+2x^2-x$ be a real-valued function. Then,the value of Lagrange's constant $C$ in $(-1,2)$ is

Let $f(x)$ be a differentiable function in $[2,7]$. If $f(2)=3$ and $f^{\prime}(x) \leq 5$ for all $x$ in $(2,7)$,then the maximum possible value of $f(x)$ at $x=7$ is

Consider the quadratic equation $ax^2+bx+c=0$,where $2a+3b+6c=0$ and let $g(x)=\frac{ax^3}{3}+\frac{bx^2}{2}+cx$.
Statement-$I$ : The given quadratic equation $ax^2+bx+c=0$ has at least one root in $(0,1)$.
Statement-$II$ : Rolle's theorem is applicable to $g(x)$ on $[0,1]$.
Then

The value of $c$ for which Rolle's theorem holds for the function $f(x)=x^3-3x^2+2x$ in the interval $[0,2]$ is:

The value $c$ of the Rolle's theorem for the function $f(x) = 2 \sin x + \sin 2x$ in the interval $[0, \pi]$ is