document.write( "Question 797678: The sum of one number and two times a second number is 24. What numbers should be selected so that their product is as large as possible? \n" ); document.write( "
Algebra.Com's Answer #481880 by solver91311(24713)\"\" \"About 
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\n" ); document.write( "\n" ); document.write( "We are interested in maximizing the product function:\r
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\n" ); document.write( "\n" ); document.write( "Which is the function we need to maximize.\r
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\n" ); document.write( "\n" ); document.write( "Algebra Way\r
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\n" ); document.write( "\n" ); document.write( "Put the quadratic function into standard form:\r
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\n" ); document.write( "\n" ); document.write( "So we have\r
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\n" ); document.write( "\n" ); document.write( "Since this is a quadratic function with a negative lead coefficient, the graph is a parabola that opens downward. Since it opens downward, the vertex is a maximum point. The -coordinate of the vertex is given by:\r
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\n" ); document.write( "\n" ); document.write( "In this case:\r
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\n" ); document.write( "\n" ); document.write( "Hence the value of for the maximum product is 6, so\r
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\n" ); document.write( "\n" ); document.write( "Calculus way\r
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\n" ); document.write( "\n" ); document.write( "Take the first derivative of the product function:\r
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\n" ); document.write( "\n" ); document.write( "Set the first derivative equal to zero and solve:\r
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\n" ); document.write( "\n" ); document.write( "Hence the function has a critical point at \r
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\n" ); document.write( "\n" ); document.write( "Take the second derivative\r
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\n" ); document.write( "\n" ); document.write( "Since the second derivative is less than zero for all in the domain of , the critical point is a maximum.\r
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\n" ); document.write( "\n" ); document.write( "Hence, the product is maximum when and \r
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\n" ); document.write( "Egw to Beta kai to Sigma
\n" ); document.write( "My calculator said it, I believe it, that settles it
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