Answer
$x_{7}=1.297383$
Work Step by Step
$f(x)=x^{5}-x^{4}+3x^{2}-3x-2$
$f`(x)=5x^4-4x^3+6x-3$
$x_{n+1}=x_{n}-\frac{x^{5}_{n}-x^{4}_{n}+3x^{2}_{n}-3x_{n}-2}{5x^4_{n}-4x^3_{n}+6x_{n}-3}$
$x_{1}=1$
$x_{2}=1-\frac{1^{5}-1^{4}+3(1)^{2}-3(1)-2}{5(1)^4-4(1)^3+6(1)-3}$
$x_{2}=1+\frac{1}{2}=1.5$
$x_{3}\approx1.5-\frac{1.5^{5}-1.5^{4}+3(1.5)^{2}-3(1.5)-2}{5(1.5)^4-4(1.5)^3+6(1.5)-3}$
$x_{3}\approx1.5-0.156140$
$x_{3}\approx1.343859$
$x_{4}\approx1.343859-0.043539$
$x_{4}\approx1.300320$
$x_{5}=1.300320-0.002924$
$x_{5}=1.297396$
$x_{6}\approx1.297396-0.000012$
$x_{6}\approx1.297383$
$x_{7}\approx1.297383-0.000795E{-7}$
$x_{7}\approx1.297383$
