document.write( "Question 1113842: The sum of 2x and y is 12. What is the largest possible product xy? \n" ); document.write( "

Algebra.Com's Answer #728914 by ikleyn(52786) $\"\"$ $\"About$

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document.write( "2x + y = 12    (given).   ====>\r\n" );
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document.write( "y = 12-2x\r\n" );
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document.write( "xy = x*(12-2x) = -2x^2 + 12x\r\n" );
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document.write( "This is an equation of the quadratic function (parabola).  Since it has negative leading coefficient at x^2, \r\n" );
document.write( "this parabola is turned downward and has a maximum.\r\n" );
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document.write( "For a general equation of a quadratic function  q(x) = ax^2 + bx + c  with a < 0,\r\n" );
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document.write( "it has a maximum at  x = .\r\n" );
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document.write( "In your case  a= -2,  b= 12  and  c= 0,  so your parabola achieves its maximum at  x=  = 3.\r\n" );
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document.write( "And the value of the maximum is the value of this quadratic function at x= 3:\r\n" );
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document.write( " = -18 + 36 = 18.\r\n" );
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document.write( "Answer.  The largest possible value of the product  xy  at given condition is 18.\r\n" );
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\n" ); document.write( "\n" ); document.write( "            Plot y = $\"-2x%5E2%2B+12x\"$ \r
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\n" ); document.write( "\n" ); document.write( "---------------
\n" ); document.write( "On finding maximum/minimum of a quadratic function see my lessons in this site \r
\n" ); document.write( "\n" ); document.write( "    - HOW TO complete the square to find the minimum/maximum of a quadratic function\r
\n" ); document.write( "\n" ); document.write( "    - Briefly on finding the minimum/maximum of a quadratic function\r
\n" ); document.write( "\n" ); document.write( "    - HOW TO complete the square to find the vertex of a parabola\r
\n" ); document.write( "\n" ); document.write( "    - Briefly on finding the vertex of a parabola\r
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\n" ); document.write( "\n" ); document.write( "    - A rectangle with a given perimeter which has the maximal area is a square\r
\n" ); document.write( "\n" ); document.write( "    - A farmer planning to fence a rectangular garden to enclose the maximal area\r
\n" ); document.write( "\n" ); document.write( "    - A farmer planning to fence a rectangular area along the river to enclose the maximal area\r
\n" ); document.write( "\n" ); document.write( "    - A rancher planning to fence two adjacent rectangular corrals to enclose the maximal area\r
\n" ); document.write( "\n" ); document.write( "    - Using quadratic functions to solve problems on maximizing revenue/profit\r
\n" ); document.write( "\n" ); document.write( "    - OVERVIEW of lessons on finding the maximum/minimum of a quadratic function\r
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\n" ); document.write( "\n" ); document.write( "Also, you have this free of charge online textbook in ALGEBRA-I in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this textbook under the topic \"Finding minimum/maximum of quadratic functions\". \r
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\n" ); document.write( "\n" ); document.write( "Save the link to this online textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-I
\n" ); document.write( "https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "to your archive and use it when it is needed.\r
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