Question

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

Answer 1

$f\left(x\right)=x+\frac{1}{x}$ $x\in [1,3]$

$\frac{x^2+1}{x}$

Ref. image

Hence,

from graph

we can see that $\begin{array}{c}\frac{x^2+1}{x}\end{array}|\begin{array}{c}x+\frac{1}{x}\end{array}$ is conf. as was differentiable

$\therefore F^{\prime }\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b=a}$

$\Rightarrow 1-\frac{1}{x^2}=\frac{3+\frac{1}{3}-2}{2}=\frac{4}{3+x}=\frac{2}{3}$

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

$\therefore x=\pm \sqrt{3}$

Hence, $x=\sqrt{x}$ which lies between $[1,3]$

$\therefore$ Its satisfies LMVT.