For which real value of $K$ does the equation $2x^3 + 3x + K = 0$ have two real roots in the interval $[0, 1]$?

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?

Let the function $f:[-7,0] \rightarrow R$ be continuous on $[-7,0]$ and differentiable on $(-7,0)$. If $f(-7)=-3$ and $f'(x) \leq 2$ for all $x \in (-7,0)$,then for all such functions $f$,$f(-1)+f(0)$ lies in the interval:

If $L.M.V.T.$ is true for $f(x) = x(x-1)(x-2)$ on the interval $x \in [0, 1/2]$,then find the value of $C$.

The constant $c$ of Rolle's theorem for the function $f(x)=(x-1)^3(x-2)^5$ in the interval $[1, 2]$ is:

In the Mean Value Theorem,$f(b) - f(a) = (b - a)f'(c)$. If $a = 4$,$b = 9$,and $f(x) = \sqrt{x}$,then the value of $c$ is: