document.write( "Question 1188322: Find two positive real numbers whose product is a maximum. (Enter your answers as a comma-separated list.)
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Algebra.Com's Answer #819381 by ikleyn(52786) $\"\"$ $\"About$

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\n" ); document.write( "Find two positive real numbers whose product is a maximum.
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\n" ); document.write( "The sum of the first and twice the second is 16.
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document.write( "Let the second number be x.\r\n" );
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document.write( "Then the first number is 16-2x.\r\n" );
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document.write( "We want to find the numbers to get the maximum value of the product  x*(16-2x).\r\n" );
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document.write( "This function,  f(x) = x*(16-2x),  is a quadratic function  f(x) = 16x - 2x^2.\r\n" );
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document.write( "Since the coefficient at x^2 is negative, its plot is a parabola opened downward, and it really has the maximum.\r\n" );
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document.write( "The zeroes of this quadratic function are x= 0 and x= 16/2 = 8,  from its decomposition f(x) = x*(16-2x).\r\n" );
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document.write( "The maximum value is located midway between the roots 0 and 8;  so, the maximum is at x= 4  and  is equal to\r\n" );
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document.write( "    4*(16-2*4) = 4*8 = 32.\r\n" );
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document.write( "ANSWER.  The maximum value of the product of the numbers is 32.\r\n" );
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document.write( "         It is achieved when second number is  4  and first number is  16 - 2*4 = 8.\r\n" );
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