Verify the Mean Value Theorem for the function $f(x) = x^{2}$ in the interval $[2, 4]$.

For a polynomial $g(x)$ with real coefficients,let $m_g$ denote the number of distinct real roots of $g(x)$. Suppose $S$ is the set of polynomials with real coefficients defined by $S = \{(x^2-1)^2(a_0+a_1x+a_2x^2+a_3x^3) : a_0, a_1, a_2, a_3 \in \mathbb{R}\}$. For a polynomial $f$,let $f'$ and $f''$ denote its first and second order derivatives,respectively. Then the minimum possible value of $(m_f + m_{f'})$,where $f \in S$,is

Examine if Rolle's Theorem is applicable to the following function. Can you say something about the converse of Rolle's Theorem from this example?
$f(x) = [x]$ for $x \in [-2, 2]$

Let $f(x) = 8x^3 - 6x^2 - 2x + 1,$ then

The value of $c$ in the Lagrange's mean value theorem for $f(x)=\sqrt{x-2}$ in the interval $[2,6]$ is

Let $f$ be a twice differentiable function on $(1,6)$. If $f(2)=8$,$f'(2)=5$,$f'(x) \geq 1$ and $f''(x) \geq 4$ for all $x \in (1,6)$,then: