document.write( "Question 907664: Among all the pairs of numbers (x,y) such that 2x-y=12, find the pair for which the sum of the squares is a minimum. What is the minimum product? \n" ); document.write( "
Algebra.Com's Answer #550492 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Notice:  I changed your word \"maximum\" to \"minimum\".\r\n" );
document.write( "That's because there is no maximum for either the sum of squares or\r\n" );
document.write( "for the product.  Surely you must have meant \"minumum\".  Otherwise\r\n" );
document.write( "there can be no solution, for the sum of the squares and the product\r\n" );
document.write( "can both be millions, billions and trillions!  They can go to \r\n" );
document.write( "infinity!!!  Here is the problem for the minimums, not maximums.\r\n" );
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document.write( "Let S = the sum of the squares of x and y\r\n" );
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document.write( "(1) S = x²+y²\r\n" );
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document.write( "Solve 2x-y = 12 for y\r\n" );
document.write( "        -y = -2x+12\r\n" );
document.write( "         y = 2x-12\r\n" );
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document.write( "Substitute for y in (1)\r\n" );
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document.write( "   S = x² + (2x-12)²\r\n" );
document.write( "   S = x² + (2x-12)(2x-12)\r\n" );
document.write( "   S = x² + (4x²-48x+144)\r\n" );
document.write( "   S = x² + 4x² - 48x + 144\r\n" );
document.write( "   S = 5x² - 48x + 144\r\n" );
document.write( "   \r\n" );
document.write( "The graph of that is below.  Since we have already used y,\r\n" );
document.write( "we have to use S, so think of the y-axis below as the S-axis.\r\n" );
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document.write( "\"graph%28400%2C400%2C-7%2C16%2C-75%2C200%2Cx%5E2%2B%282x-12%29%5E2%29\"\r\n" );
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document.write( "From that graph you can see there can be no maximum.  However\r\n" );
document.write( "there is a minimum at the vertex.\r\n" );
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document.write( "Using the vertex formula:\r\n" );
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document.write( "The x-coordinate of the vertex = \"-b%2F%282a%29=-%28-48%29%2F%282%285%29%29=4.8\"\r\n" );
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document.write( "The S-coordinate of the vertex is S = 5(4.8)² - 48(4.8) + 144 = 28.8\r\n" );
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document.write( "Now we must find y when x = 4.8\r\n" );
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document.write( "               y = 2(4.8)-12\r\n" );
document.write( "               y = -2.4\r\n" );
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document.write( "So the pair (x,y) for which the sum of the squares is a minimum is (4.8,-2.4)\r\n" );
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document.write( "For the minimum product:\r\n" );
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document.write( "Let P = the product of x and y\r\n" );
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document.write( "(1) P = xy\r\n" );
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document.write( "We've already solved 2x-y = 12 for y\r\n" );
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document.write( "         y = 2x-12\r\n" );
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document.write( "Substitute for y in (1)\r\n" );
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document.write( "   P = x(2x-12)\r\n" );
document.write( "   P = 2x²-12x\r\n" );
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document.write( "The graph of that is below.  Since we have already used y,\r\n" );
document.write( "we have to use P, so think of the y-axis below as the P-axis.\r\n" );
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document.write( "\"graph%28400%2C400%2C-7%2C16%2C-75%2C200%2C2x%5E2-12x%29\"\r\n" );
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document.write( "Just as the other one, from that graph you can see there can \r\n" );
document.write( "be no maximum.  However there is a minimum at the vertex.\r\n" );
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document.write( "Using the vertex formula again:\r\n" );
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document.write( "The x-coordinate of the vertex = \"-b%2F%282a%29=-%28-12%29%2F%282%282%29%29=3\"\r\n" );
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document.write( "The P-coordinate of the vertex is P = 2(3)²-12(3) = -18\r\n" );
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document.write( "Now we must find y when x = 3\r\n" );
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document.write( "y = 2(3)-12\r\n" );
document.write( "               y = -6\r\n" );
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document.write( "So the pair (x,y) for which the product is a minimum is (3,-6).\r\n" );
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document.write( "Edwin

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