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

If a polynomial equation $a_nx^n + a_{n-1}x^{n-1} + \dots + a_2x^2 + a_1x + a_0 = 0$,where $n$ is a positive integer,has two distinct roots $\alpha$ and $\beta$,then how many roots does the equation $na_nx^{n-1} + (n - 1)a_{n-1}x^{n-2} + \dots + a_1 = 0$ have in the interval $(\alpha, \beta)$?

The value of $c$ satisfying the conditions and conclusions of Rolle's theorem for the function $f(x) = x \sqrt{x+6}$ on the interval $x \in [-6, 0]$ is:

Suppose $f(x)$ is twice differentiable in the interval $[1, 3]$ and $f(1)=f(3)$. If $|f^{\prime \prime}(x)| \leq 2$,then for all $x$ in $[1, 3]$,which one of the following is true?

If the $L.M.V.T.$ holds for the function $f(x) = x + \frac{1}{x}$ on the interval $x \in [1, 3]$,then $c$ is:

Let $f(1) = -2$ and $f'(x) \ge 4.2$ for $1 \le x \le 6$. The smallest possible value of $f(6)$ is: