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document.write( "Find two nonnegative numbers whose sum is 9 and so that the product of one
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document.write( "number and the square of the other number is a maximum 10.
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document.write( " Formulation of your post is NOT CORRECT and makes a few sense, if any . . . \r
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document.write( " Meanwhile, an interesting problem may stay behind it --- so I edited it, to make sense and to create really interesting problem.\r
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document.write( " My editing is as follows.\r
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document.write( " Prove that if the sum of two non-negative real number \"x\" and \"y\" is 9, x + y = 9,\r\n" );
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document.write( " then the product 
has the local maximum value of 108.\r\n" );
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document.write( " Find the values of x and y that provide this maximum.\r\n" );
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document.write( " With this reformulation, the solution is below.\r
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document.write( "Solution\r
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document.write( "If x+y = 9, then y = 9-x, and the function f(x,y) = is\r\n" );
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document.write( " f(x,y) = = 
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document.write( "So, we need to find the maximum of the function of x g(x) = .\r\n" );
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document.write( "To find the maximum of the function g(x), take its derivative and equate it to zero.\r\n" );
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document.write( "The derivative is\r\n" );
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document.write( " g'(x) = 
- 2x*(9-x) = 
- 
= 
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document.write( "Equating it to zero, you get\r\n" );
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= 0, which is equivalent to 
= 0.\r\n" );
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document.write( "Left side is factorable\r\n" );
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document.write( " (x-9)*(x-3) = 0.\r\n" );
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document.write( "and two solution of the quadratic equation are x= 9 and x= 3.\r\n" );
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document.write( "The value x= 9 provides the local minimum of the function f(x), while x= 3 provides the local maximum.\r\n" );
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document.write( "At x= 3, y= 9-3 = 6, and the function g(x,y) = 
= 108.\r\n" );
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document.write( " 
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document.write( " Plot g(x) = x*(9-x)^2\r\n" );
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document.write( "Solved.\r
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