Answer
$x_{3}=1.5215$
Work Step by Step
$f(x)=\frac{2}{x}-x^{2}+1$
$f′(x)=\frac{-2}{x^{2}}-2x$
$x_{1}=2$
$x_{n+1}=x_{n}-\frac{\frac{2}{x_{n}}-x^{2}_{n}+1}{\frac{-2}{x^{2}_{n}}-2x_{n}}$
$x_{2}=2-\frac{\frac{2}{(2)}-(2)^{2}+1}{\frac{-2}{(2)^{2}}-2(2)}$
$x_{2}=2-\frac{-2}{-4.5}$
$x_{2}=\frac{14}{9}$
$x_{3}=\frac{14}{9}-\frac{\frac{2}{(\frac{14}{9})}-(\frac{14}{9})^{2}+1}{\frac{-2}{(\frac{14}{9})^{2}}-2(\frac{14}{9})}$
$x_{3}=\frac{14}{9}-0.0340$
$x_{3}=1.5215$
