Let f: S → S where S = (0, ∞) be a twice differentiable function...

lnf\left(x + 1\right) = \ln \left(xf\left(x\right)\right)$</p><p>$lnf\left(x + 1\right) = \ln x + lnf \left(x\right)$</p><p>$⇒ g\left(x + 1\right) = \ln x + g\left(x\right)$</p><p>$⇒ g\left(x + 1\right) - g ( x ) = \ln x$</p><p>$⇒ {g}^{''}\left(x + 1\right) - {g}^{''}\left(x\right) = - \dfrac{1}{{x}^{2}}$</p><p>Put$x = 1, 2, 3, 4$</p><p>${g}^{''}\left(2\right) - {g}^{''}\left(1\right) = - \dfrac{1}{{1}^{2}} . . . \left(1\right)$</p><p>${g}^{''}\left(3\right) - {g}^{''}\left(2\right) = - \dfrac{1}{{2}^{2}} . . . \left(2\right)$</p><p>${g}^{''}\left(4\right) - {g}^{''}\left(3\right) = - \dfrac{1}{{3}^{2}} . . . \left(3\right)$</p><p>${g}^{''}\left(5\right) - {g}^{''}\left(4\right) = - \dfrac{1}{{4}^{2}} . . . \left(4\right)$</p><p>Add all the equations we get</p><p>${g}^{''}\left(5\right) - {g}^{''}\left(1\right) = - \dfrac{1}{{1}^{2}} - \dfrac{1}{{2}^{2}} - \dfrac{1}{{3}^{2}} - \dfrac{1}{{4}^{2}}$</p><p>\left|{g}^{"}\left(5\right) - {g}^{"}\left(1\right)\right| = \dfrac{205}{144}$