Science:Math Exam Resources/Courses/MATH102/December 2017/Question 11 (b)/Solution 1

Recall from the solution for part (a) that $$f'(x)=5x^{4}-8x+2,$$ $$f''(x)=20x^{3}-8.$$

We need to solve ${\textstyle f'(x)=0}$ near ${\textstyle x_{0}=1}$. Using the Hint with ${\textstyle f'(x)}$ in place of ${\textstyle F(x)}$, we have $$x_{1}=x_{0}-{\dfrac {f'(x_{0})}{f''(x_{0})}}=1-{\dfrac {f'(1)}{f''(1)}}=1-{\dfrac {5-8+2}{20-8}}=1-{\dfrac {-1}{12}}={\dfrac {13}{12}}.$$

Answer: the next approximation is ${\textstyle {\color {blue}x_{1}={\dfrac {13}{12}}}}$.