Sat Dec 04, 2010 6:10 pm by tartle 


write 271 as the sum of positive real numbers so as to maximize their product. 




Mon Jan 03, 2011 1:13 pm by s.b. 


2+3+4+5+6+7+8....
note:0 is not applied cuz then product becomes 0. 1 does not change the product...am i right? 




Sun May 29, 2011 7:48 pm by bds021 


135.5+135.5 




Sat Jun 04, 2011 7:03 am by Unni 


3+2+2+2+2+2.......
Product = 3 x 134th power of 2 




Sat Jun 04, 2011 7:05 am by Unni 


3+2+2+2+2+2.......
Product = 3 x 134th power of 2 




Sun Jun 19, 2011 6:05 pm by DiamondSoul 


It's 2.71 repeated 100 times. 




Wed Jul 06, 2011 4:28 pm by cat 


[quote="DiamondSoul"]It's 2.71 repeated 100 times.[/quote]
This appears to be correct, but needs a little explanation.
([i]n[/i]+1)*([i]n[/i]1) = [i]n[/i]^21, which is less than [i]n[/i]^2. This shows that uniform values adding to a given sum make the largest product. Therefore, using [i]a[/i] for 271, the product [i]y[/i] of [i]x[/i] uniform values can be written as:
[i]y[/i] = ([i]a[/i]/[i]x[/i])^[i]x[/i] = e^[[i]x[/i]*(ln[i]a[/i]ln[i]x[/i])]
The derivative is:
[i]dy[/i]/[i]dx[/i] = [([i]a[/i]/[i]x[/i])^[i]x[/i]]*[ln[i]a[/i](1+ln[i]x[/i])] = [([i]a[/i]/[i]x[/i])^[i]x[/i]]*[ln([i]a[/i]/[i]x[/i])1]
At the maximum product, the derivative equals zero, so that:
ln([i]a[/i]/[i]x[/i]) = 1;
[i]a[/i]/[i]x[/i] = e;
[i]x[/i] = [i]a[/i]/e = 271/2.718... = 100 to the nearest integer. 






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