Let

Question:

Let $\mathrm{f}: \mathrm{S} \rightarrow \mathrm{S}$ where $\mathrm{S}=(0, \infty)$ be a twice differentiable function such that $\mathrm{f}(\mathrm{x}+1)=\mathrm{xf}(\mathrm{x})$ If $g: S \rightarrow R$ be defined as $g(x)=\log _{e} f(x)$, then the value of $\left|g^{\prime \prime}(5)-\mathrm{g}^{\prime \prime}(1)\right|$ is equal to :

1. (1) $\frac{205}{144}$

2. (2) $\frac{197}{144}$

3. (3) $\frac{187}{144}$

4. (4) 1

Correct Option: 1

Solution:

$\operatorname{lnf}(x+1)=\ln (x f(x))$

$\operatorname{lnf}(x+1)=\ln x+\operatorname{lnf}(x)$

$\Rightarrow g(x+1)=\ln x+g(x)$

$\Rightarrow g(x+1)-g(x)=\ln x$

$\Rightarrow \quad g^{\prime \prime}(x+1)-g^{\prime \prime}(x)=-\frac{1}{x^{2}}$

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

$\mathrm{g}^{\prime \prime}(2)-\mathrm{g}^{\prime \prime}(1)=-\frac{1}{1^{2}} \quad \ldots(1)$

$\mathrm{g}^{\prime \prime}(3)-\mathrm{g}^{\prime \prime}(2)=-\frac{1}{2^{2}} \quad \ldots(2)$

$\mathrm{g}^{\prime \prime}(4)-\mathrm{g}^{\prime \prime}(3)=-\frac{1}{3^{2}}$

$g^{\prime \prime}(5)-g^{\prime \prime}(4)=-\frac{1}{4^{2}} \quad \ldots(4)$

Add all the equation we get

$\mathrm{g}^{\prime \prime}(5)-\mathrm{g}^{\prime \prime}(1)=-\frac{1}{1^{2}}-\frac{1}{2^{2}}-\frac{1}{3^{2}}-\frac{1}{4^{2}}$

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