Algebra.Com's Answer #840923 by ikleyn(52802)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "A company has $33600 to spend on the development and promotion of a new product.
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document.write( "The company estimates that if x is spent on development and y is spent on promotion,
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document.write( "then approximately ((x^1/2)(y^3/2))/400,000 items of new product will be sold.
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document.write( "Based on this estimate, how much should the company spend on development
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document.write( "so the maximum number of products will be sold?
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document.write( " It is clear that this problem is on finding maximum.\r
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document.write( " It can be solved by applying standard Calculus to the given function,
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document.write( " but it is, obviously, not very pleasant exersize, taking into account
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document.write( " complicated formula for the function.\r
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document.write( " There is a trick, and I will show it below, which makes the solution an easy and pleasant
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document.write( " procedure and makes a reader happy, since he (or she) learns something new.\r
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document.write( " Solution\r\n" );
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document.write( "Let u = (x^(1/2))*(y^(3/2))/400000 be the given function;\r\n" );
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document.write( " v = (x^(1/2))*(y^(3/2));\r\n" );
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document.write( " w =  .\r\n" );
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document.write( "It is obvious that \"u\" and \"v\" have the maximum at the same point (x,y)\r\n" );
document.write( "(dividing by the constant value 400,000 does not shift the position of maximum on (x,y)-plane).\r\n" );
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document.write( "Which is even more interesting and productive, is the fact that \"v\" and \"w\" also have \r\n" );
document.write( "the maximum at the same point (x,y) on the coordinate plane.\r\n" );
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document.write( " | It is because w(x,y) = v^2(x,y). |\r\n" );
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document.write( "Now our task is much more simple: find the point (x,y) on the coordinate plane such that\r\n" );
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document.write( " x + y = 33600 (1)\r\n" );
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document.write( "and \r\n" );
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document.write( " maximizes w(x,y) = , x >=0, y >= 0. (2)\r\n" );
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document.write( "To do it, from (1) I express y = 33600-x and substitute it into (2).\r\n" );
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document.write( "Then I get the problem to maximize \r\n" );
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document.write( " W(x) =  .\r\n" );
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document.write( "Apply a standard Calculus procedure: find the derivative and equate it to zero.\r\n" );
document.write( "You will get then\r\n" );
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document.write( " (33600-x)^3 =  , 0 < x < 33600.\r\n" );
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document.write( "Simplify by reducing the factor  in both sides\r\n" );
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document.write( " 33600-x = 3x\r\n" );
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document.write( " 33600 = 3x + x = 4x\r\n" );
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document.write( " x = 33600/4 = 8400.\r\n" );
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document.write( "So, the maximum of w(x,y), v(x,y) and u(x,y) is achieved at x= 8400, y= 33600-8400 = 25200\r\n" );
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document.write( " max u(x,y) = u(8400,25200) = (8400^(1/2)*25200^(3/2)))/400000 = 916.6 (rounded)\r\n" );
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document.write( "ANSWER. The point of maximum is (x,y) = (8400,25200).\r\n" );
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document.write( " The value of the maximum is about 916.6.\r\n" );
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document.write( " The amount to spend on development is x = $8400, leaving $25200 to spend on promotion.\r\n" );
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document.write( "Solved.\r
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document.write( "From my solution learn this trick, which makes the solution easy.\r
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