If $x = \alpha$ is a positive root of the equation $a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x = 0$,then what can be said about the positive root of the equation $na_nx^{n-1} + (n-1)a_{n-1}x^{n-2} + \dots + a_1 = 0$?

The function $f(x) = x(x + 3)e^{-\frac{1}{2}x}$ satisfies all the conditions of Rolle's theorem in $[-3, 0]$. Find the value of $c$ such that $f'(c) = 0$.

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

Examine if Rolle's Theorem is applicable to the function $f(x) = [x]$ for $x \in [5, 9]$. Can you say something about the converse of Rolle's Theorem from this example?

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

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