## DBsubject('Calculus')
## DBchapter('Differentiation in Several Variables')
## DBsection('Optimization in Several Variables')
## KEYWORDS('calculus')
## TitleText1('Calculus: Early Transcendentals')
## EditionText1('2')
## AuthorText1('Rogawski')
## Section1('14.7')
## Problem1('7')
## Author('JustAsk - Vladimir Finkelshtein')
## Institution('W.H.Freeman')

DOCUMENT();

loadMacros("PG.pl","PGbasicmacros.pl","PGanswermacros.pl");
loadMacros("Parser.pl");
loadMacros("freemanMacros.pl");
loadMacros("PGauxiliaryFunctions.pl");
loadMacros("PGgraphmacros.pl");
loadMacros("PGchoicemacros.pl");

TEXT(beginproblem());

do{
  $a=non_zero_random(-4,4,1);
} while($a == 2 || $a == -2);
$ma = Real(-$a);

$x0=non_zero_random(-8,8,1);
$b=$x0*(($a)**2/2-2);

Context()->texStrings;
$context = Context();
$context->variables->add(y=>'Real');

$f=Formula("x^2+y^2+$a*x*y+$b*x")->reduce();
$fx=Formula("2*x+$a*y+$b")->reduce();
$fy=Formula("2*y+$a*x")->reduce();

$y=Formula("-$a/2*x")->reduce();
$ynr=Formula("-$a/2*x");
$ma2=Formula("-$a/2");
$y0=$y->eval(x=>$x0);

$fxx=2;
$fyy=2;
$fxy=$a;

$disc=$fxx*$fyy-($fxy)**2;

if($disc>0) {$ans='a local minimum';
              $wrongans1='test fails';
              $wrongans2='a local maximum'; 
              $wrongans3='a saddle point';
              $sol="\quad f_{xx}($x0,$y0)=2>0";
              $sign='>'; }
if($disc<0) {$ans='a saddle point';
              $wrongans1='test fails';
              $wrongans2='a local maximum'; 
              $wrongans3='a local minimum'; 
              $sign='<'; }

$mc = new_multiple_choice();
$mc -> qa ("","$ans",);
$mc ->extra("$wrongans1","$wrongans2","$wrongans3",);

BEGIN_TEXT
\{ textbook_ref_exact("Rogawski ET 2e", "14.7","7") \}
$PAR
Find the critical point of the function \(f(x,y)=$f\)
$PAR
\(c=\) \{ans_rule()\}
$PAR
Use the Second Derivative Test to determine whether the point is
\{$mc -> print_a\}
$BR
END_TEXT 

Context()->normalStrings;
Context("Vector");
ANS(Point(Real($x0),Formula($y0))->cmp);
Context("Numeric");
ANS(str_cmp($mc->correct_ans));
Context()->texStrings;

SOLUTION(EV3(<<'END_SOLUTION'));
$PAR
$SOL
$BR
First, we find the critical point. $BR
We set the first-order partial derivatives of \(f(x,y)=$f\) equal to zero and solve
\[f_x(x,y)=$fx=0\]
\[f_y(x,y)=$fy=0\]
The second equation implies that \(y=$ynr\). Substituting into the first and solving for \(x\) gives:
\[2x+$ynr+$b=0 \Rightarrow x=$x0\]
The corresponding value of \(y\) is \(y=$ma2($x0)=$y0\). The critical point is \(($x0,$y0)\).
$PAR
Now we compute the Discriminant. We find the second-order partials:
\[f_{xx}(x,y)=2 \quad f_{yy}(x,y)=2 \quad f_{xy}(x,y)=$a\]
The discriminant is 
\[D(x,y)=f_{xx}f_{yy}-f_{xy}^2=2\cdot 2-($a)^2=$disc\]
$PAR
Applying the Second Derivative Test we have
\[D($x0,$y0)=$disc $sign 0 $sol\]
The Second Derivative test implies that \(f($x0,$y0)\) is $ans.
$BR
END_SOLUTION

ENDDOCUMENT();