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)![]() ![]() You can put this solution on YOUR website! \r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( " \n" ); document.write( " \n" ); document.write( "\n" ); document.write( " \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "We are interested in maximizing the product function:\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( " \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "Substituting:\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( " \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "Which is the function we need to maximize.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "Algebra Way\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "Put the quadratic function into standard form:\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( " \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "So we have\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( " \n" ); document.write( " \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 \n" ); document.write( " \n" ); document.write( "\n" ); document.write( " \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "In this case:\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( " \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "Hence the value of \n" ); document.write( " \n" ); document.write( "\n" ); document.write( " \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "Calculus way\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "Take the first derivative of the product function:\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( " \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "Set the first derivative equal to zero and solve:\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( " \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "Hence the function has a critical point at \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "Take the second derivative\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( " \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "Since the second derivative is less than zero for all \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "Hence, the product is maximum when \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "John \n" ); document.write( " \n" ); document.write( "Egw to Beta kai to Sigma \n" ); document.write( "My calculator said it, I believe it, that settles it \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " |