Let f : S → S where S = 0 , ∞ be a twice differentiable function such that f x + 1 = x f x . If…

$lnf\left(\mathrm{x}+1\right)=ln\left(xf\left(x\right)\right)$

$lnf\left(x+1\right)=lnx+lnf\left(x\right)$

$\Rightarrow g\left(x+1\right)=lnx+g\left(x\right)$

$\Rightarrow g\left(x+1\right)-g\left(x\right)=lnx$

$\Rightarrow g^{\text{'}\text{'}}\left(x+1\right)-g^{\text{'}\text{'}}\left(x\right)=-\frac{1}{x^2}$

Put $x=1,2,3,4$

$g^{\text{'}\text{'}}\left(2\right)-g^{\text{'}\text{'}}\left(1\right)=-\frac{1}{1^2} ...\left(1\right)$

$g^{\text{'}\text{'}}\left(3\right)-g^{\text{'}\text{'}}\left(2\right)=-\frac{1}{2^2} ...\left(2\right)$

$g^{\text{'}\text{'}}\left(4\right)-g^{\text{'}\text{'}}\left(3\right)=-\frac{1}{3^2} ...\left(3\right)$

$g^{\text{'}\text{'}}\left(5\right)-g^{\text{'}\text{'}}\left(4\right)=-\frac{1}{4^2} ...\left(4\right)$

Add all the equations we get

$g^{\text{'}\text{'}}\left(5\right)-g^{\text{'}\text{'}}\left(1\right)=-\frac{1}{1^2}-\frac{1}{2^2}-\frac{1}{3^2}-\frac{1}{4^2}$

$\left|g^{"}\left(5\right)-g^{"}\left(1\right)\right|=\frac{205}{144}$