Question

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

Answer 1

$f\left(x\right)=\frac{x+1}{x}$

a=1, b-3.

For LMVT, all three conditions verify

therefore Let x=c be a number c= (1,3)

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

$f^{\prime }\left(x\right)=\frac{x\left(d/dx\right(x+1\left)\right)-(x+1)d/dx}{x^2}$

=> $\frac{x-(x+1)}{x^2}$
=> $\frac{-1}{x^2}$

$f^{\prime }\left(c\right)=\frac{-1}{c^2}$

=>$\frac{f\left(3\right)-f\left(1\right)}{3-1}$

=>$\frac{-2}{6}$

Therefore, $c=+/-\sqrt{3}$