View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/14_Differentiation_in_Several_Variables/14.7_Optimization_in_Several_Variables/14.7.37.pg

    1 ## DBsubject('Calculus')
    2 ## DBchapter('Differentiation in Several Variables')
    3 ## DBsection('Optimization in Several Variables')
    4 ## KEYWORDS('calculus')
    5 ## TitleText1('Calculus: Early Transcendentals')
    6 ## EditionText1('2')
    7 ## AuthorText1('Rogawski')
    8 ## Section1('14.7')
    9 ## Problem1('37')
   10 ## Author('JustAsk - Vladimir Finkelshtein')
   11 ## Institution('W.H.Freeman')
   12 ## UsesAuxiliaryFiles('14-7-35-image.png')
   13 
   14 DOCUMENT();
   15 
   16 loadMacros("PG.pl","PGbasicmacros.pl","PGanswermacros.pl");
   17 loadMacros("Parser.pl");
   18 loadMacros("freemanMacros.pl");
   19 loadMacros("PGauxiliaryFunctions.pl");
   20 loadMacros("PGgraphmacros.pl");
   21 loadMacros("PGchoicemacros.pl");
   22 
   23 TEXT(beginproblem());
   24 
   25 Context()->texStrings;
   26 $context = Context();
   27 $context->variables->add(y=>'Real');
   28 $a=random(1,5,1);
   29 $b=random(2,8,1);
   30 $aminb=$a-$b;
   31 
   32 $f=Formula("$a*x^3-$b*y")->reduce();
   33 $fx=Formula("3*$a*x^2")->reduce();
   34 $fy=Real(-$b);
   35 
   36 $f1=Formula("$a*x^3")->reduce();
   37 $f2=Formula("$a-$b*y")->reduce();
   38 $f3=Formula("$a*x^3-$b")->reduce();
   39 $f4=Formula("-$b*y")->reduce();
   40 
   41 $fmin=-$b;
   42 $fmax=$a;
   43 
   44 BEGIN_TEXT
   45 \{ textbook_ref_exact("Rogawski ET 2e", "14.7","37") \}
   46 $PAR
   47 Determine the global extreme values of the function
   48 \(f(x,y)=$f, \quad 0 \le x,y \le 1\)
   49 $PAR
   50 \(f_{min}=\)\{ans_rule()\} $BR
   51 \(f_{max}=\)\{ans_rule()\}
   52 $BR
   53 END_TEXT
   54 
   55 Context()->normalStrings;
   56 ANS(Real($fmin)->cmp);
   57 ANS(Real($fmax)->cmp);
   58 Context()->texStrings;
   59 
   60 SOLUTION(EV3(<<'END_SOLUTION'));
   61 $PAR
   62 $SOL
   63 $BR
   64 First we find the critical points
   65 \[f_x(x,y)=$fx=0, \quad f_y(x,y)=$fy\]
   66 The two equations have no solutions, hence there are no critical points.
   67 $BR
   68 The extreme values occur either at the critical points or at a point on the boundary of the domain. $BR
   69 To check the boundary we consider each edge of the square \(0\le x,y \le 1\) separately.$BR
   70 \{image("14-7-35-image.png", width=>150, height=>150)\}
   71 $BR
   72 The segment \(\mathrm{OA}\): On this segment \(y=0, 0\le x\le 1\), and \(f\) takes the values \(f(x,0)=$f1\). $BR
   73 The minimum value is \(f(0,0)=0\) and the maximum value is \(f(1,0)=$a\). $PAR
   74 The segment \(\mathrm{AB}\): On this segment \(x=1, 0\le y\le 1\), and \(f\) takes the values \(f(1,y)=$f2\). $BR
   75 The minimum value is \(f(1,1)=$aminb\) and the maximum value is \(f(1,0)=$a\). $PAR
   76 The segment \(\mathrm{BC}\): On this segment \(y=1, 0\le x\le 1\), and \(f\) takes the values \(f(x,1)=$f3\). $BR
   77 The minimum value is \(f(0,1)=-$b\) and the maximum value is \(f(1,1)=$aminb\). $PAR
   78 The segment \(\mathrm{OC}\): On this segment \(x=0, 0\le y\le 1\), and \(f\) takes the values \(f(0,y)=$f4\). $BR
   79 The minimum value is \(f(0,1)=-$b\) and the maximum value is \(f(0,0)=0\). $PAR
   80 
   81 The smallest value is \(f_{min}=$fmin\) and it is the global minimum of \(f\) on the square. $BR
   82 The global maximum is the largest value \(f_{max}=$fmax\).
   83 $BR
   84 END_SOLUTION
   85 
   86 ENDDOCUMENT();