(* finding local minima for a function of one variable *) 


 F[x_] := x^4 - 2x^2 + 0.1x; 

(* F[x_] := x^20; *)    

(* F[x_] := x^2 + Sin[30*x];  *)  

(* F[x_] := x^2 * Exp[-x^2]; *) 


(* answer = FindMinimum [F[x], {x,  1.5 } ] *)


answer = FindMinimum [F[x], {x, 0.3}, Method -> Newton, AccuracyGoal -> 5,  MaxIterations -> 3 ]

xmin =  x /. answer[ [2] ] 
ymin = answer[ [1] ]

Print["  "]
Print["Value of X that minimizes F, xmin =   ", xmin]
Print["  "]
Print["                          F(xmin)=     ", ymin]
Print[" "]

gr1 = Plot[F[x], {x, -2, 2}];

gr2 = Plot[ 100*(x - xmin), {x, xmin - 0.01, xmin + 0.01}];

Show[gr1,gr2]