(* Content-type: application/mathematica *)

(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)

(* CreatedBy='Mathematica 7.0' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[       145,          7]
NotebookDataLength[      5053,        186]
NotebookOptionsPosition[      4359,        157]
NotebookOutlinePosition[      4702,        172]
CellTagsIndexPosition[      4659,        169]
WindowFrame->Normal*)

(* Beginning of Notebook Content *)
Notebook[{
Cell[BoxData[
 RowBox[{
  RowBox[{"f", "[", "k_", "]"}], ":=", 
  RowBox[{
   RowBox[{"k", " ", 
    FractionBox[
     SuperscriptBox["N", "2"], "2"]}], "-", " ", 
   RowBox[{"k", " ", 
    FractionBox[
     SuperscriptBox["s", "2"], 
     RowBox[{"2", " ", "b", " ", 
      RowBox[{"(", 
       RowBox[{"k", "+", "b"}], ")"}]}]]}]}]}]], "Input",
 CellChangeTimes->{{3.481014688232627*^9, 3.4810147452489815`*^9}, {
  3.4810377351940746`*^9, 3.4810377355378313`*^9}}],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"D", "[", 
   RowBox[{
    RowBox[{"f", "[", "k", "]"}], ",", "k"}], "]"}], "//", 
  "Simplify"}]], "Input",
 CellChangeTimes->{{3.4810147471865063`*^9, 3.481014753545963*^9}}],

Cell[BoxData[
 FractionBox[
  RowBox[{
   RowBox[{
    SuperscriptBox["b", "2"], " ", 
    SuperscriptBox["N", "2"]}], "+", 
   RowBox[{"2", " ", "b", " ", "k", " ", 
    SuperscriptBox["N", "2"]}], "+", 
   RowBox[{
    SuperscriptBox["k", "2"], " ", 
    SuperscriptBox["N", "2"]}], "-", 
   SuperscriptBox["s", "2"]}], 
  RowBox[{"2", " ", 
   SuperscriptBox[
    RowBox[{"(", 
     RowBox[{"b", "+", "k"}], ")"}], "2"]}]]], "Output",
 CellChangeTimes->{3.48101475410847*^9, 3.4810361478174644`*^9, 
  3.481037738537889*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"kRoots", "=", 
  RowBox[{"Solve", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"D", "[", 
      RowBox[{
       RowBox[{"f", "[", "k", "]"}], ",", "k"}], "]"}], "\[Equal]", "0"}], 
    ",", "k"}], "]"}]}]], "Input",
 CellChangeTimes->{{3.4810147567647543`*^9, 3.481014768655531*^9}, {
  3.4810149125792484`*^9, 3.4810149138448896`*^9}}],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"k", "\[Rule]", 
     FractionBox[
      RowBox[{
       RowBox[{
        RowBox[{"-", "b"}], " ", "N"}], "-", "s"}], "N"]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"k", "\[Rule]", 
     FractionBox[
      RowBox[{
       RowBox[{
        RowBox[{"-", "b"}], " ", "N"}], "+", "s"}], "N"]}], "}"}]}], 
  "}"}]], "Output",
 CellChangeTimes->{
  3.4810147692336636`*^9, 3.4810149143761463`*^9, 3.481036170615069*^9, 
   3.4810365599244013`*^9, {3.4810377403660493`*^9, 3.4810377495381002`*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"specificRoots", "=", 
  RowBox[{
   RowBox[{"kRoots", "/.", 
    RowBox[{"{", 
     RowBox[{
      RowBox[{"s", "\[Rule]", "1"}], ",", 
      RowBox[{"b", "\[Rule]", "1"}], ",", 
      RowBox[{"N", "\[Rule]", 
       FractionBox["1", "2"]}]}], "}"}]}], "//", "Simplify"}]}]], "Input",
 CellChangeTimes->{{3.4810149152667828`*^9, 3.481014927079434*^9}, {
  3.481014968064334*^9, 3.4810150113305845`*^9}, {3.481037830727159*^9, 
  3.481037833492837*^9}}],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"k", "\[Rule]", 
     RowBox[{"-", "3"}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"k", "\[Rule]", "1"}], "}"}]}], "}"}]], "Output",
 CellChangeTimes->{
  3.481014972955021*^9, {3.4810150051117153`*^9, 3.481015011814969*^9}, 
   3.4810366216291313`*^9, 3.481037750991253*^9, 3.4810378338678446`*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"D", "[", 
    RowBox[{
     RowBox[{"f", "[", "k", "]"}], ",", 
     RowBox[{"{", 
      RowBox[{"k", ",", "2"}], "}"}]}], "]"}], "/.", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"s", "\[Rule]", "1"}], ",", 
     RowBox[{"b", "\[Rule]", "1"}], ",", 
     RowBox[{"N", "\[Rule]", 
      FractionBox["1", "2"]}]}], "}"}]}], "/.", "specificRoots"}]], "Input",
 CellChangeTimes->{{3.481037818320671*^9, 3.4810378555870113`*^9}}],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
   RowBox[{"-", 
    FractionBox["1", "8"]}], ",", 
   FractionBox["1", "8"]}], "}"}]], "Output",
 CellChangeTimes->{{3.4810378232113895`*^9, 3.481037856305775*^9}}]
}, Open  ]]
},
WindowSize->{616, 750},
WindowMargins->{{0, Automatic}, {Automatic, 0}},
FrontEndVersion->"7.0 for Microsoft Windows (32-bit) (February 18, 2009)",
StyleDefinitions->"Default.nb"
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[545, 20, 467, 14, 52, "Input"],
Cell[CellGroupData[{
Cell[1037, 38, 210, 6, 31, "Input"],
Cell[1250, 46, 527, 17, 52, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[1814, 68, 359, 10, 31, "Input"],
Cell[2176, 80, 563, 18, 47, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[2776, 103, 476, 12, 47, "Input"],
Cell[3255, 117, 371, 10, 30, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[3663, 132, 472, 14, 47, "Input"],
Cell[4138, 148, 205, 6, 47, "Output"]
}, Open  ]]
}
]
*)

(* End of internal cache information *)