Applications of Derivatives — Maths STD 12 Science — Question

step 1: Given
$f(x)=x+\frac{1}{x}, x \in[1,3]$
We know that a polynomial function is continuous everywhere and also differentiable. So, 
being a polynomial is continuous and differentiable on (1,3), So there must exist at least one real number c ∈ [1, 3]  such that
$f^{\prime}(c)=\frac{f(3)-f(1)}{3-1}$
step2:
$f(x)=x+\frac{1}{x}$
$f(3)=\frac{10}{3}$
$f(1)=2$
$f^{\prime}(x)=1-\frac{1}{x^2}$
$f^{\prime}(c)=1-\frac{1}{c^2}$
$1-\frac{1}{c^2}=\frac{\frac{10}{3}-2}{2}$
$\frac{c^2-1}{c^2}=\frac{2}{3}$
$3 c^2-3=2 c^2$
$c^2=3$
$c=\sqrt{3}$
$c \in(1,3)$
Hence Lagrange's Mean Value theorem is verified