Newton's Method and graphing

Programming

Clear[NaiveNewton]
Off[
      General::spell1]
NaiveNewton[f_, guess_]:=
	Module[{newtstep, root},
		newtstep = 1;
		root = guess;
		While[Abs[newtstep] > 10^(-6),
			newtstep=N[-((f[root])/(f'[root]))];
			root = root + newtstep;
		Print[root];
		];
		N[root, 6]]

Problem:  graph    f[x] := (x^4) - 17 (x^2) + 3 x - 5

Clear[f]
f[x_] := (x^4) - 17 (x^2) + 3 x - 5
g[x_]  =  f'[x]
h[x_]  =  g'[x]

We would like to know where 
g(x)=f'(x)=4 (x^3) - 34 x + 3 = 0.   
g  is a cubic polynomial,  
      so it has either 1 or 3 zeros.

\(\(\(
\)\(Plot[g[x], \ {x, \ \(-10\), \ 10}, \ 
      Axes -> False]\)\)\)

If we find out where the local maximum and local minimum are, then we can also find out whether g has one or three zeros, and where they are, approximately.  Then we can use Newton's method to locate the zeros more exactly.
The local maximum and minimum occur where 
h(x)=g'(x)=12 (x^2) - 34 = 0.
That's something we can finally solve by hand; the zeros of h are at  ± (Sqrt[17/12]), so that's where the local max and min of g are.
Let's evaluate g[x] at these values of x.

N[g[-(Sqrt[17/12])]]
N[g[(Sqrt[17/12])]]

Since the local max is positive and the local min negative, there is one zero of g[x] in between the two,  one zero to the left of the max and one zero to the right of the min.

α  =  NaiveNewton[g,  -3]
β  =  
      NaiveNewton[g,  0]

γ  =  NaiveNewton[g,  3]

g[x] = f'[x]  is negative to the left of   α,  positive between   α and  β, negative between   β   and  γ, and positive to the right of   γ. 
 So f is decreasing  to the left of  α , increasing between   α and  β, decreasing between   β   and  γ, and increasing to the right of   γ.

\(
f[α]\)
f[β]
f[γ]

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

What about concavity?  The graph of f is concave up when 
f''[x]  = h[x] = 12 (x^2) - 34
is positive, and concave down when f''[x] is negative.
So the graph of f is concave up to the left of  -(Sqrt[17/12]) and concave down to the right of (Sqrt[17/12]).

In summary, the important points on the graph of  y = f[x] are at 
x = ± (Sqrt[17/12]) and at x =

α

β

γ