Lagrange's Mean Value Theorem

The value of $\text{'}c\text{'},$ of the Lagrange's mean value theorem, for $f\left(x\right)=\sqrt{x^2-x}, x\in [1,4]$ is

A function $f\left(x\right)$ fulfil the conditions of Lagrange's mean value theorem in $\left[0,5\right]$. If $\left|f^{\text{'}}\left(x\right)\right|\le \frac{1}{4} \forall x\in \left[0,5\right]$ & $f\left(0\right)=0$, then which of the following is true

The $c$ value  of Lagrange’s mean-value theorem for $f\left(x\right)=\sqrt{25-x^2}$ on $\left[1,5\right]$ is $\sqrt{k}$. Find $k$.

Function $f\left(x\right)=2x^2-3x+1$ does not satisfies Lagrange’s mean-value theorem on $\left[1,3\right]$.

Lagrange’s mean-value theorem is not applicable on function $f\left(x\right)=\frac{1}{x}$ on $\left[-1,1\right]$.

Function $f\left(x\right)=x^2+2x+3$ does not satisfies Lagrange’s mean-value theorem on $\left[4,6\right]$.

Function $f\left(x\right)=x^3+x^2-6x$ satisfies Lagrange’s mean-value theorem on $\left[-1,4\right]$.

The $‘c’$ value  of Lagrange’s mean-value theorem for $\mathrm{f}\left(\mathrm{x}\right)=\sqrt{\mathrm{x}+2} \mathrm{on} \left[4,6\right]$ is $\frac{k}{2}+2\sqrt{k}$. Find $k$.

The $c$ value  of Lagrange’s mean-value theorem for $f\left(x\right)=x^3-3x^2+2x$ on $\left[0,\frac{1}{2}\right]$ is equal to $1-\sqrt{\frac{k}{12}}$. Find $k$.

Function $f\left(x\right)=x^{\frac{2}{3}}$ on $\left[0,1\right]$ satisfies Lagrange’s mean-value theorem.

Function $f\left(x\right)=e^x$ on $\left[0,1\right]$ does not satisfies Lagrange’s mean-value theorem.

Function $f\left(x\right)=\log x$ on $\left[1,e\right]$ satisfies Lagrange's mean-value theorem.

Verify L.M.V.T. for the function

$f\left(x\right)=x{\left(x+4\right)}^2, x\in \left[0,4\right]$

Find $c$ if LMVT is applicable for
$f\left(x\right)=x(x-1)(x-2), x\in [0,1/2]$

Verify LMVT for the following function

$f\left(x\right)=x^2-3x-1,x\in \left[-\frac{11}{7},\frac{13}{7}\right]$

Verify LMVT for the following function
$f\left(x\right)=x+\frac{1}{x}, x\in [1,3]$

Verify $\mathrm{LMVT}$ for the following function.
 $f\left(x\right)=x(2-x),x\in [0,1]$

Verify Lagrange's mean value theorem for the following function 

$f\left(x\right)=\left\{\begin{array}{l}1+x\\ 5-x\end{array}\right.\begin{array}{ll};& x<2\\ ;& x\ge 2\end{array};x\in [1,3]$

Verify Lagrange's mean value theorem for the following function $f\left(x\right)=(x-1)(x-2)(x-3); x\in [0,4]$

Verify Lagrange's mean value theorem for the function $f\left(x\right)=x^2-3x+2; x\in [-2,3]$