F[x_] = (x^3)-5 (x^2)+3 x + 5;

RootsF = NSolve[F[x]==0,x]

RootsPlot = Graphics[{PointSize[0.025],Red,Point[{x,0}/.RootsF]}];

PlotF = Plot[F[x],{x,-3,7},PlotStyle->{Blue, Thickness[0.007]}];

Show[PlotF,RootsPlot,PlotRange->{{-2,5},{-5,6}}, AspectRatio->1]

TangentF1 = Plot[-2.437865843178432520232911283494022992540080629291`37.237622569891414( x -0.68424321924104989329905295977669962995)+5.032140742481771614934756817887214588901507875073`37.65071614043538,{x,0,3},PlotStyle->{Red, Thickness[0.007]}];

TangentF2 = Plot[-1.82288318366929137634756859473`10.398590021649941( x -2.74840140766744176642003632328`12.)-3.76271925011573825381111395012`10.410958056904322,{x,0,3},PlotStyle->{Black, Thickness[0.007]}];

Show[TangentF2,TangentF1,PlotF,RootsPlot,PlotRange->{{-2,5},{-5,6}}, AspectRatio->1]

DF[2.74840140766744176642003632328`12.]

F[2.74840140766744176642003632328`12.]

DF[x_] = D[F[x],x]

N[Solve[DF[x]==0,x]]

Newt[x_] = x-F[x]/DF[x]

(N[NestList[Newt[#]&,0.68424321924104989329905295977669962995,20],12])//MatrixForm

(N[NestList[Newt[#]&,0.69,10],12])//MatrixForm

(N[NestList[Newt[#]&,0.67,10],12])//MatrixForm

(N[NestList[Newt[#]&,2.749,10],12])//MatrixForm

(N[NestList[Newt[#]&,2.746,10],12])//MatrixForm

(N[NestList[Newt[#]&,1/3+0.0000001,50],12])

(N[NestList[Newt[#]&,0.200000000001,10],12])//MatrixForm

(N[NestList[Newt[#]&,-1,6],12])//MatrixForm

(N[NestList[Newt[#]&,1,5],12])//MatrixForm

(N[NestList[Newt[#]&,2,5],12])//MatrixForm

(N[NestList[Newt[#]&,5,8],12])//MatrixForm

(N[NestList[Newt[#]&,0,8],12])//MatrixForm

N[Solve[Newt[x]==x,x]]

N[Solve[Newt[Newt[x]]==x,x, WorkingPrecision->20],20]

N[Solve[Newt[Newt[Newt[x]]]==x,x, WorkingPrecision->20],20]

Newt4[x_] = Simplify[Newt[Newt[Newt[Newt[x]]]]];

Plot[{Newt4[x],x},{x,0.4,0.8},PlotStyle->{{Red},{Blue}}, PlotPoints->500, ClippingStyle->None,PerformanceGoal->"Quality"]

(Subscript[x, 0]) = 0.68424321924104989329905295977669962995

(N[NestList[Newt[#]&,(Subscript[x, 0]),12],12])//MatrixForm

SNewt[x_] = Simplify[((D[Newt[x],{x,3}])/(D[Newt[x],x]))-(3/2) (((((D[Newt[x],{x,2}])/(D[Newt[x],x]))))^2)]

Plot[SNewt[x],{x,-5,5}]