Verify MVT hypotheses and find the guaranteed value of c

First find the average rate of change: $f(1) = 1 + 1 = 2$ and $f(3) = 3 + \tfrac{1}{3} = \tfrac{10}{3}$, so $\dfrac{f(3)-f(1)}{3-1} = \dfrac{10/3 - 2}{2} = \dfrac{4/3}{2} = \dfrac{2}{3}$. This is the slope of the secant line joining the endpoints $(1,2)$ and $(3,10/3)$.

Next compute $f'(x) = 1 - \dfrac{1}{x^2}$ and set it equal to $\tfrac{2}{3}$: $1 - \dfrac{1}{x^2} = \tfrac{2}{3}$ gives $\dfrac{1}{x^2} = \tfrac{1}{3}$, so $x^2 = 3$ and $x = \pm\sqrt{3}$. Since $c$ must lie in $(1,3)$, we take $c = \sqrt{3}$.

Geometrically, $f'(\sqrt{3}) = \tfrac{2}{3}$ means the tangent line to $f$ at $x = \sqrt{3}$ has the same slope as the secant line through $(1, f(1))$ and $(3, f(3))$ — that is, the tangent line at $c$ is parallel to the secant line connecting the endpoints of the curve on $[1,3]$.