Let $f(1) = -2$ and $f'(x) \ge 4.2$ for $1 \le x \le 6$. The possible value of $f(6)$ lies in the interval

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be such that $f(0)=0$ and $|f^{\prime}(x)| \leq 5$ for all $x$. Then $f(1)$ is in

Given that $f(x)$ is continuously differentiable on $a \le x \le b$ where $a < b, f(a) < 0$ and $f(b) > 0$,which of the following are always true?
$(i)$ $f(x)$ is bounded on $a \le x \le b$.
$(ii)$ The equation $f(x) = 0$ has at least one solution in $a < x < b$.
$(iii)$ The maximum and minimum values of $f(x)$ on $a \le x \le b$ occur at points where $f'(c) = 0$.
$(iv)$ There is at least one point $c$ with $a < c < b$ where $f'(c) > 0$.
$(v)$ There is at least one point $d$ with $a < d < b$ where $f'(d) < 0$.

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

If $f:[a, b] \rightarrow [c, d]$ is a continuous and strictly increasing function, then $\frac{d-c}{b-a}$ is

Consider the following functions:
$I) f(x) = \begin{cases} \frac{1}{2}-x, & x < \frac{1}{2} \\ (\frac{1}{2}-x)^2, & x \geq \frac{1}{2} \end{cases}$
$II) f(x) = |3x-1|$
$III) f(x) = x|x|$
$IV) f(x) = |x|$
Then on $[0, 1]$,Lagrange's Mean Value Theorem $(LMVT)$ is applicable to which of the functions?