For the function $f(x) = x + \frac{1}{x},x \in [1,3],$ the value of $c$ for mean value theorem is

$\Rightarrow$ $1 - \frac{1}{{{c^2}}} = \frac{{\left[ {3 + \frac{1}{3}} \right] - \left[ {1 + \frac{1}{1}} \right]}}{{3 - 1}}$

$\Rightarrow$ $\frac{{{c^2} - 1}}{{{c^2}}} = \frac{{\frac{{10}}{3} - 2}}{2}$

$\Rightarrow$ $\frac{{{c^2} - 1}}{{{c^2}}} = \frac{4}{{3 \times 2}} = \frac{2}{3}$

$\Rightarrow$ $3\left( {{c^2} - 1} \right) = 2{c^2}$

$\Rightarrow$ $3{c^2} - 2{c^2} = 3$

$\Rightarrow$ ${c^2} = 3 \Rightarrow c = \pm \sqrt 3$