Discuss the applicability of Lagrange's mean value theorem to the function $f\left(x\right)=\frac{1}{x}$ in $[-1, 1]$.

Discuss the applicability of Lagrange's mean value theorem to the function $f\left(x\right)=4-(6-x{)}^{\frac{2}{3}}$ in $[5, 7]$.

$f\left(x\right)=\left\{\begin{array}{l}x \sin \frac{1}{x},\text{ for }x\ne 0\\ 0 ,\text{ for }x=0\end{array}\right.$ in $[-1, 1]$.

Find the value of $c$ that appears in the conclusion of Lagrange's mean value theorem in $f\left(x\right)=\log x$ in $[1, e]$.