given f(x)= x^(1/4)-4x^(3)+10x^(2)-12x+7, find taylor series of f(x) about x=1. Show that the series is finite

Question

also use the answer to the above to show that f(x) >= 2 for all x

Answer 1

By definition a series has an infinite number of terms and also with x^1/4 the Taylor expression won´t have a finite number of thermes
If
f(x) = x^4-4x^3+10x^2 -12x+7
f(x) =f(1) +(x-1)f´(1) +(x-1)^2*f´´(1)/2!+(x-1)^3*f´´´(1)/3!+(x-1)^4*f¨¨(1)/4!
so f(x) =2+ 8*(x-1)^2/2!+24(x-1)^4/4!
as f´(1)=f´´´(1)=0 and al derivatives of order>4 are 0
All terms have even indexes so they are >= 0 so f(x)>=2