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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...

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.

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Answer #1

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

for critical points f '(x)=0
=>x2-2=0

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

using newton's method

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