## DESCRIPTION
##   Max/min problems
## ENDDESCRIPTION

## KEYWORDS('Critical', 'Point', 'Partial', 'Multivariable')
## Tagged by nhamblet

## DBsubject('Calculus')
## DBchapter('Partial Derivatives')
## DBsection('Maximum and Minimum Values')
## Date('October 29, 2009')
## Author('Shifrin')
## Institution('UGA')
## TitleText1('')
## EditionText1('')
## AuthorText1('')
## Section1('')
## Problem1('')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
  "PGstandard.pl",
  "PGunion.pl",
  "Parser.pl",
  "parserVectorUtils.pl",
  "PGcourse.pl"
);


TEXT(beginproblem());
BEGIN_PROBLEM();

##############################################
#  Setup

Context("Vector");

$a = random(1,4);
do{$b = non_zero_random(-3,3)} until ($b!=1);
do{$c = non_zero_random(-4,4)} until ($c**2<4*$a);

$f = Formula("x^2 + $b y^2 + $c x")->reduce;
$arg = '\left(\begin{array}{c} x\\y \end{array}\right)';


$max = max($c**2/(4*($b-1))+ $a*$b, $a+sqrt($a)*$c, $a-sqrt($a)*$c, -$c**2/4);
$min = min($c**2/(4*($b-1))+ $a*$b, $a+sqrt($a)*$c, $a-sqrt($a)*$c, -$c**2/4);


##############################################
#  Main text

Context()->texStrings;
BEGIN_TEXT

The temperature of the circular plate \(D = \left\{"\{"\} $arg: x^2+y^2 \le $a \right\}\) is given by the function \(f $arg=$f\). Find the maximum and minimum temperatures of \(D\). 

$PAR
maximum temperature = \{ans_rule(10)\}
$PAR
minimum temperature = \{ans_rule(10)\}

$PAR

END_TEXT
Context()->normalStrings;

##################################################
#  Answers

ANS(num_cmp($max));
ANS(num_cmp($min));

$showPartialCorrectAnswers = 1;

##################################################

END_PROBLEM();
ENDDOCUMENT();        # This should be the last executable line in the problem.