(* Content-type: application/mathematica *)

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

(* CreatedBy='Mathematica 6.0' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[       145,          7]
NotebookDataLength[     16602,        575]
NotebookOptionsPosition[     14190,        503]
NotebookOutlinePosition[     14556,        519]
CellTagsIndexPosition[     14513,        516]
WindowFrame->Normal*)

(* Beginning of Notebook Content *)
Notebook[{
Cell["Extreme Values and Saddle Points", "Title"],

Cell["", "Subsubtitle",
 CellChangeTimes->{{3.45536418275*^9, 3.45536418346875*^9}}],

Cell["Stuff you should know", "Subsubsection"],

Cell[TextData[{
 "Recall:\n1.) f(a,b) is a ",
 StyleBox["local maximum ",
  FontWeight->"Bold"],
 "value of f if f(a,b) >= f(x,y) for all ",
 StyleBox["domain",
  FontSlant->"Italic"],
 " points (x,y) in an open disk centered at (a,b).\n2",
 ".) f(a,b) is a ",
 StyleBox["local minimum ",
  FontWeight->"Bold"],
 "value of f if f(a,b) <= f(x,y) for all ",
 StyleBox["domain",
  FontSlant->"Italic"],
 " points (x,y) in an open disk centered at (a,b).",
 "\n3.) An interior point of the domain of a function where f_x and f_y are \
both zero ( \[Del]f=0 ), or where one or both of f_x and f_y do not exist is \
called a ",
 StyleBox["critical point",
  FontWeight->"Bold"]
}], "Text"],

Cell["Example: Finding Local Extrema", "Subsubsection"],

Cell["\<\
Consider the function below, on the domain 0 \[LessEqual] x \[LessEqual] 2 , \
-1 \[LessEqual] y \[LessEqual] 1\
\>", "Text"],

Cell[BoxData[
 RowBox[{
  RowBox[{"f", "[", 
   RowBox[{"x_", ",", "y_"}], "]"}], "=", 
  RowBox[{
   RowBox[{"6", "x"}], "-", 
   RowBox[{"2", 
    RowBox[{"x", "^", "2"}]}], "-", 
   RowBox[{"3", "x", "*", 
    RowBox[{"y", "^", "3"}]}], "+", 
   RowBox[{
    RowBox[{"x", "^", "2"}], 
    RowBox[{"y", "^", "3"}]}], "+", 
   RowBox[{"4.5", "*", "x", "*", 
    RowBox[{"y", "^", "4"}]}], "-", 
   RowBox[{"1.5", "*", 
    RowBox[{"x", "^", "2"}], "*", 
    RowBox[{"y", "^", "4"}]}], "+", 
   RowBox[{"12", "x", "*", 
    RowBox[{"Sin", "[", 
     RowBox[{"Pi", "*", "y"}], "]"}]}], "-", 
   RowBox[{"4", 
    RowBox[{"x", "^", "2"}], "*", 
    RowBox[{"Sin", "[", 
     RowBox[{"Pi", "*", "y"}], "]"}]}], "+", 
   RowBox[{"4", "x", "*", "y"}]}]}]], "Input"],

Cell[TextData[{
 "Our task is to find the ",
 StyleBox["local extrema ",
  FontWeight->"Bold"],
 "of this function, that is, find and classify all interior critical points \
on the domain"
}], "Text"],

Cell["Let's make a plot!", "Text"],

Cell[BoxData[{
 RowBox[{"p", "=", 
  RowBox[{"Plot3D", "[", 
   RowBox[{
    RowBox[{"f", "[", 
     RowBox[{"x", ",", "y"}], "]"}], ",", 
    RowBox[{"{", 
     RowBox[{"x", ",", "0", ",", "2"}], "}"}], ",", 
    RowBox[{"{", 
     RowBox[{"y", ",", 
      RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", 
    RowBox[{"PlotRange", "\[Rule]", "All"}], ",", 
    RowBox[{"PlotPoints", "\[Rule]", "30"}], ",", 
    RowBox[{"AxesLabel", "\[Rule]", 
     RowBox[{"{", 
      RowBox[{"\"\<x\>\"", ",", "\"\<y\>\"", ",", "\"\<f(x,y)\>\""}], 
      "}"}]}]}], "]"}]}], "\[IndentingNewLine]", 
 RowBox[{"Show", "[", 
  RowBox[{"p", ",", 
   RowBox[{"ViewPoint", "\[Rule]", 
    RowBox[{"{", 
     RowBox[{"2", ",", 
      RowBox[{"-", "0.6"}], ",", "0.5"}], "}"}]}]}], "]"}]}], "Input"],

Cell["Lets also make a contour plot", "Text"],

Cell[BoxData[{
 RowBox[{"ContourPlot", "[", 
  RowBox[{
   RowBox[{"f", "[", 
    RowBox[{"x", ",", "y"}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"x", ",", "0", ",", "2"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"y", ",", 
     RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", 
   RowBox[{"Contours", "\[Rule]", "30"}], ",", 
   RowBox[{"PlotPoints", "\[Rule]", "75"}], ",", 
   RowBox[{"FrameLabel", "\[Rule]", 
    RowBox[{"{", 
     RowBox[{"\"\<x\>\"", ",", "\"\<y\>\""}], "}"}]}]}], 
  "]"}], "\[IndentingNewLine]", 
 RowBox[{"ContourPlot", "[", 
  RowBox[{
   RowBox[{"f", "[", 
    RowBox[{"x", ",", "y"}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"x", ",", "0", ",", "2"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"y", ",", 
     RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", 
   RowBox[{"Contours", "\[Rule]", "100"}], ",", 
   RowBox[{"PlotPoints", "\[Rule]", "100"}], ",", 
   RowBox[{"ContourLines", "\[Rule]", "False"}], ",", 
   RowBox[{"FrameLabel", "\[Rule]", 
    RowBox[{"{", 
     RowBox[{"\"\<x\>\"", ",", "\"\<y\>\""}], "}"}]}]}], "]"}]}], "Input"],

Cell["\<\
We have critical points where \[Del] f  = 0, so lets make a function for the \
gradient...\
\>", "Text"],

Cell[BoxData[
 RowBox[{
  RowBox[{"gradf", "[", 
   RowBox[{"x_", ",", "y_"}], "]"}], "=", 
  RowBox[{"{", 
   RowBox[{
    RowBox[{"D", "[", 
     RowBox[{
      RowBox[{"f", "[", 
       RowBox[{"x", ",", "y"}], "]"}], ",", "x"}], "]"}], " ", ",", " ", 
    RowBox[{"D", "[", 
     RowBox[{
      RowBox[{"f", "[", 
       RowBox[{"x", ",", "y"}], "]"}], ",", "y"}], "]"}]}], "}"}]}]], "Input"],

Cell["\<\
...but the command Solve[] will not work...(try it if you don't believe me)\
\>", "Text"],

Cell["\<\
So, lets try something different.  We are going to use FindRoot[ ] but, it \
needs an initial guess near the critical point.  Lets make plots in the x-y \
plane where df/dx=0, and where df/dy=0.  Where these curves intersect, we \
have critical points!\
\>", "Text"],

Cell[BoxData[
 RowBox[{"p1", "=", 
  RowBox[{"ContourPlot", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"gradf", "[", 
       RowBox[{"x", ",", "y"}], "]"}], "[", 
      RowBox[{"[", "1", "]"}], "]"}], "\[Equal]", "0"}], ",", 
    RowBox[{"{", 
     RowBox[{"x", ",", "0", ",", "2"}], "}"}], ",", 
    RowBox[{"{", 
     RowBox[{"y", ",", 
      RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", 
    RowBox[{"ColorFunction", "\[Rule]", 
     RowBox[{"Function", "[", 
      RowBox[{"Hue", "[", "1", "]"}], "]"}]}]}], "]"}]}]], "Input",
 CellChangeTimes->{{3.454520231921875*^9, 3.454520240390625*^9}, {
  3.455364209921875*^9, 3.455364236234375*^9}, {3.4553643690625*^9, 
  3.45536440015625*^9}}],

Cell[BoxData[
 RowBox[{"p2", "=", 
  RowBox[{"ContourPlot", "[", 
   RowBox[{
    RowBox[{
     RowBox[{
      RowBox[{"gradf", "[", 
       RowBox[{"x", ",", "y"}], "]"}], "[", 
      RowBox[{"[", "2", "]"}], "]"}], "\[Equal]", "0"}], ",", 
    RowBox[{"{", 
     RowBox[{"x", ",", "0", ",", "2"}], "}"}], ",", 
    RowBox[{"{", 
     RowBox[{"y", ",", 
      RowBox[{"-", "1"}], ",", "1"}], "}"}]}], "]"}]}]], "Input",
 CellChangeTimes->{{3.455364241546875*^9, 3.455364242359375*^9}, {
  3.455364407015625*^9, 3.455364408109375*^9}}],

Cell[BoxData[
 RowBox[{"Show", "[", 
  RowBox[{"p1", ",", "p2"}], "]"}]], "Input"],

Cell["\<\
Looks like somewhere around ( 1.6 , 0.6 ) we have a critical points.  We also \
could have guessed this from the contour plot!
Since the blue and red lines don't intersect anywhere else in the domain, \
that is the only critical point!\
\>", "Text",
 CellChangeTimes->{{3.455364418015625*^9, 3.455364419109375*^9}}],

Cell[BoxData[
 RowBox[{"sol", "=", 
  RowBox[{"FindRoot", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"gradf", "[", 
      RowBox[{"x", ",", "y"}], "]"}], "\[Equal]", 
     RowBox[{"{", 
      RowBox[{"0", ",", "0"}], "}"}]}], ",", 
    RowBox[{"{", 
     RowBox[{"x", ",", "1.6"}], "}"}], ",", 
    RowBox[{"{", 
     RowBox[{"y", ",", "0.6"}], "}"}]}], "]"}]}]], "Input"],

Cell["\<\
We can use the command \"/.\" to assign the values found above to a handle.\
\>", "Text",
 CellChangeTimes->{{3.454520328109375*^9, 3.454520384140625*^9}, {
  3.454520497875*^9, 3.45452050459375*^9}}],

Cell[BoxData[{
 RowBox[{"x1", "=", 
  RowBox[{"x", "/.", 
   RowBox[{"sol", "[", 
    RowBox[{"[", "1", "]"}], "]"}]}]}], "\[IndentingNewLine]", 
 RowBox[{"y1", "=", 
  RowBox[{"y", "/.", 
   RowBox[{"sol", "[", 
    RowBox[{"[", "2", "]"}], "]"}]}]}]}], "Input"],

Cell["Lets check what this is using the 2nd derivative test", "Text"],

Cell[BoxData[{
 RowBox[{
  RowBox[{"fxx", "[", 
   RowBox[{"x_", ",", "y_"}], "]"}], "=", 
  RowBox[{"D", "[", 
   RowBox[{
    RowBox[{"f", "[", 
     RowBox[{"x", ",", "y"}], "]"}], ",", 
    RowBox[{"{", 
     RowBox[{"x", ",", "2"}], "}"}]}], "]"}]}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"fyy", "[", 
   RowBox[{"x_", ",", "y_"}], "]"}], "=", 
  RowBox[{"D", "[", 
   RowBox[{
    RowBox[{"f", "[", 
     RowBox[{"x", ",", "y"}], "]"}], ",", 
    RowBox[{"{", 
     RowBox[{"y", ",", "2"}], "}"}]}], "]"}]}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"fxy", "[", 
   RowBox[{"x_", ",", "y_"}], "]"}], "=", 
  RowBox[{"D", "[", 
   RowBox[{
    RowBox[{"D", "[", 
     RowBox[{
      RowBox[{"f", "[", 
       RowBox[{"x", ",", "y"}], "]"}], ",", "x"}], "]"}], ",", "y"}], 
   "]"}]}]}], "Input"],

Cell[BoxData[
 RowBox[{
  RowBox[{"M", "[", 
   RowBox[{"x_", ",", "y_"}], "]"}], "=", 
  RowBox[{"{", 
   RowBox[{
    RowBox[{"{", 
     RowBox[{
      RowBox[{"fxx", "[", 
       RowBox[{"x", ",", "y"}], "]"}], ",", 
      RowBox[{"fxy", "[", 
       RowBox[{"x", ",", "y"}], "]"}]}], "}"}], ",", 
    RowBox[{"{", 
     RowBox[{
      RowBox[{"fxy", "[", 
       RowBox[{"x", ",", "y"}], "]"}], ",", 
      RowBox[{"fyy", "[", 
       RowBox[{"x", ",", "y"}], "]"}]}], "}"}]}], "}"}]}]], "Input"],

Cell[BoxData[
 RowBox[{"%", "//", "MatrixForm"}]], "Input"],

Cell[BoxData[
 RowBox[{"fxx", "[", 
  RowBox[{"x1", ",", "y1"}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"Det", "[", 
  RowBox[{"M", "[", 
   RowBox[{"x1", ",", "y1"}], "]"}], "]"}]], "Input"],

Cell["\<\
fxx < 0 and Det[ M ] = fxx*fyy - fxy^2 > 0 --> We have a local max, the value \
of the function here is\
\>", "Text"],

Cell[BoxData[
 RowBox[{"f", "[", 
  RowBox[{"x1", ",", "y1"}], "]"}]], "Input"],

Cell["\<\
Let's plot f(x,y) around this critical point...it sure does look like a max! \
yay!\
\>", "Text"],

Cell[BoxData[
 RowBox[{"Plot3D", "[", 
  RowBox[{
   RowBox[{"f", "[", 
    RowBox[{"x", ",", "y"}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"x", ",", 
     RowBox[{"x1", "-", "0.2"}], ",", 
     RowBox[{"x1", "+", "0.2"}]}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"y", ",", 
     RowBox[{"y1", "-", "0.1"}], ",", 
     RowBox[{"y1", "+", "0.1"}]}], "}"}], ",", 
   RowBox[{"AxesLabel", "\[Rule]", 
    RowBox[{"{", 
     RowBox[{"\"\<x\>\"", ",", "\"\<y\>\"", ",", "\"\<f(x,y)\>\""}], 
     "}"}]}]}], "]"}]], "Input"],

Cell["\<\
Example: Finding Global Extrema and  the Boundary of the Domain\
\>", "Subsubsection"],

Cell["\<\
Since our domain is a rectangle, we have 4 sides to consider.  Any of the 4 \
sides could have the global max / global min.\
\>", "Text"],

Cell["\<\
Consider first the 'right' side of the domain, where x=2.  On this edge, our \
function f(x,y) is really only a function of ONE variable, g( y ) = f ( 2 , y \
)\
\>", "Text"],

Cell[BoxData[
 RowBox[{
  RowBox[{"g", "[", "y_", "]"}], "=", 
  RowBox[{"f", "[", 
   RowBox[{"2", ",", "y"}], "]"}]}]], "Input"],

Cell[BoxData[
 RowBox[{"Plot", "[", 
  RowBox[{
   RowBox[{"g", "[", "y", "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"y", ",", 
     RowBox[{"-", "1"}], ",", "1"}], "}"}]}], "]"}]], "Input"],

Cell["\<\
Looks like this function has a max and a min!  Let's use calc I to find them\
\>", "Text"],

Cell[BoxData[
 RowBox[{
  RowBox[{"gy", "[", "y_", "]"}], "=", 
  RowBox[{"D", "[", 
   RowBox[{
    RowBox[{"g", "[", "y", "]"}], ",", "y"}], "]"}]}]], "Input"],

Cell[BoxData[
 RowBox[{"sol", "=", 
  RowBox[{"FindRoot", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"gy", "[", "y", "]"}], "\[Equal]", "0"}], ",", 
    RowBox[{"{", 
     RowBox[{"y", ",", "0.5"}], "}"}]}], "]"}]}]], "Input"],

Cell[BoxData[
 RowBox[{"y2", "=", 
  RowBox[{"y", "/.", "sol"}]}]], "Input"],

Cell[BoxData[
 RowBox[{"x2", "=", "2"}]], "Input"],

Cell[BoxData[
 RowBox[{"sol", "=", 
  RowBox[{"FindRoot", "[", 
   RowBox[{
    RowBox[{
     RowBox[{"gy", "[", "y", "]"}], "\[Equal]", "0"}], ",", 
    RowBox[{"{", 
     RowBox[{"y", ",", 
      RowBox[{"-", "0.5"}]}], "}"}]}], "]"}]}]], "Input"],

Cell[BoxData[
 RowBox[{"y3", "=", 
  RowBox[{"y", "/.", "sol"}]}]], "Input"],

Cell[BoxData[
 RowBox[{"x3", "=", "2"}]], "Input"],

Cell[BoxData[
 RowBox[{"f", "[", 
  RowBox[{"x2", ",", "y2"}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"f", "[", 
  RowBox[{"x3", ",", "y3"}], "]"}]], "Input"],

Cell[CellGroupData[{

Cell["The point ( 0.609 , 2 )", "Section"],

Cell["\<\
Lets look at the full 3D function around these two points.  Be careful not to \
plot anything outside of the domain!\
\>", "Text"],

Cell[BoxData[
 RowBox[{"Plot3D", "[", 
  RowBox[{
   RowBox[{"f", "[", 
    RowBox[{"x", ",", "y"}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"x", ",", 
     RowBox[{"x2", "-", "0.2"}], ",", "x2"}], "}"}], ",", 
   RowBox[{"{", 
    RowBox[{"y", ",", 
     RowBox[{"y2", "-", "0.1"}], ",", 
     RowBox[{"y2", "+", "0.1"}]}], "}"}], ",", 
   RowBox[{"AxesLabel", "\[Rule]", 
    RowBox[{"{", 
     RowBox[{"\"\<x\>\"", ",", "\"\<y\>\"", ",", "\"\<f(x,y)\>\""}], 
     "}"}]}]}], "]"}]], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["The point ( -0.55 , 2 )", "Section"],

Cell[BoxData[
 RowBox[{"p3", "=", 
  RowBox[{"Plot3D", "[", 
   RowBox[{
    RowBox[{"f", "[", 
     RowBox[{"x", ",", "y"}], "]"}], ",", 
    RowBox[{"{", 
     RowBox[{"x", ",", 
      RowBox[{"x3", "-", "0.3"}], ",", "x3"}], "}"}], ",", 
    RowBox[{"{", 
     RowBox[{"y", ",", 
      RowBox[{"y3", "-", "0.1"}], ",", 
      RowBox[{"y3", "+", "0.1"}]}], "}"}], ",", 
    RowBox[{"AxesLabel", "\[Rule]", 
     RowBox[{"{", 
      RowBox[{"\"\<x\>\"", ",", "\"\<y\>\"", ",", "\"\<f(x,y)\>\""}], 
      "}"}]}]}], "]"}]}]], "Input"],

Cell[BoxData[
 RowBox[{"Show", "[", 
  RowBox[{"p3", ",", 
   RowBox[{"ViewPoint", "->", 
    RowBox[{"{", 
     RowBox[{"1.924", ",", " ", 
      RowBox[{"-", "2.112"}], ",", " ", "0.726"}], "}"}]}]}], "]"}]], "Input"],

Cell["\<\
You should examine the other three edges of the boundary!!!  Determine all \
local and absolute max's and min's.  Are there any saddles? Rotate the image \
to gain more insight.\
\>", "Text",
 CellChangeTimes->{{3.4545205539375*^9, 3.454520607515625*^9}}]
}, Open  ]]
},
WindowSize->{1272, 877},
WindowMargins->{{0, Automatic}, {Automatic, 0}},
ShowSelection->True,
FrontEndVersion->"7.0 for Microsoft Windows (32-bit) (February 18, 2009)",
StyleDefinitions->"DemoText.nb"
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[545, 20, 49, 0, 56, "Title"],
Cell[597, 22, 84, 1, 56, "Subsubtitle"],
Cell[684, 25, 46, 0, 44, "Subsubsection"],
Cell[733, 27, 683, 20, 115, "Text"],
Cell[1419, 49, 55, 0, 44, "Subsubsection"],
Cell[1477, 51, 135, 3, 25, "Text"],
Cell[1615, 56, 760, 25, 38, "Input"],
Cell[2378, 83, 200, 6, 25, "Text"],
Cell[2581, 91, 34, 0, 25, "Text"],
Cell[2618, 93, 774, 22, 59, "Input"],
Cell[3395, 117, 45, 0, 25, "Text"],
Cell[3443, 119, 1071, 30, 59, "Input"],
Cell[4517, 151, 114, 3, 25, "Text"],
Cell[4634, 156, 396, 13, 38, "Input"],
Cell[5033, 171, 99, 2, 25, "Text"],
Cell[5135, 175, 276, 5, 43, "Text"],
Cell[5414, 182, 706, 19, 38, "Input"],
Cell[6123, 203, 535, 15, 38, "Input"],
Cell[6661, 220, 82, 2, 38, "Input"],
Cell[6746, 224, 325, 6, 55, "Text"],
Cell[7074, 232, 373, 12, 38, "Input"],
Cell[7450, 246, 210, 4, 25, "Text"],
Cell[7663, 252, 263, 8, 59, "Input"],
Cell[7929, 262, 69, 0, 25, "Text"],
Cell[8001, 264, 809, 28, 79, "Input"],
Cell[8813, 294, 500, 17, 38, "Input"],
Cell[9316, 313, 59, 1, 38, "Input"],
Cell[9378, 316, 81, 2, 38, "Input"],
Cell[9462, 320, 110, 3, 38, "Input"],
Cell[9575, 325, 127, 3, 25, "Text"],
Cell[9705, 330, 79, 2, 38, "Input"],
Cell[9787, 334, 107, 3, 25, "Text"],
Cell[9897, 339, 524, 16, 38, "Input"],
Cell[10424, 357, 96, 2, 44, "Subsubsection"],
Cell[10523, 361, 147, 3, 25, "Text"],
Cell[10673, 366, 184, 4, 25, "Text"],
Cell[10860, 372, 130, 4, 38, "Input"],
Cell[10993, 378, 189, 6, 38, "Input"],
Cell[11185, 386, 100, 2, 25, "Text"],
Cell[11288, 390, 161, 5, 38, "Input"],
Cell[11452, 397, 227, 7, 38, "Input"],
Cell[11682, 406, 76, 2, 38, "Input"],
Cell[11761, 410, 50, 1, 38, "Input"],
Cell[11814, 413, 249, 8, 38, "Input"],
Cell[12066, 423, 76, 2, 38, "Input"],
Cell[12145, 427, 50, 1, 38, "Input"],
Cell[12198, 430, 79, 2, 38, "Input"],
Cell[12280, 434, 79, 2, 38, "Input"],
Cell[CellGroupData[{
Cell[12384, 440, 42, 0, 52, "Section"],
Cell[12429, 442, 140, 3, 25, "Text"],
Cell[12572, 447, 496, 15, 38, "Input"]
}, Open  ]],
Cell[CellGroupData[{
Cell[13105, 467, 42, 0, 52, "Section"],
Cell[13150, 469, 534, 16, 38, "Input"],
Cell[13687, 487, 219, 6, 38, "Input"],
Cell[13909, 495, 265, 5, 25, "Text"]
}, Open  ]]
}
]
*)

(* End of internal cache information *)