Question

Use Newton’s Method to approximate a critical number of the function ?(?)=(1/3)?^3−2?+6. f(x)=1/3x^3−2x+6 near the point ?=1x=1. Find the next two approximations, ?2 and ?3 using ?1=1. x1=1 as the initial approximation.

Answer 1

given f(x)=(1_/3)x³-2x+6
=>f '(x)=(1_/3)(3)x^3-1-2(1)+0
=>f '(x)=x²​​​​​​​-2

for critical points f '(x)=0
=>x²-2=0

let p(x)=x²-2
=>p'(x)=2x^2-1-0
=>p'(x)=2x

using newton's method