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Optimal result (Posted on 2018-06-04) Difficulty: 2 of 5
A stick of length L is broken into n equal parts.

What is the maximal product of their lengths?

No Solution Yet Submitted by Ady TZIDON    
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re(2): What Ady wants (spoiler?) | Comment 5 of 11 |
(In reply to re: What Ady wants (spoiler?) by Ady TZIDON)

this is the approach I had used as well


d/dx ln(y) = d/dx [n*ln(L)-n*ln(n)]
now we want to know when y'=0
since L>0 and n>=1 we know y>0 thus we need

as was already determined and as was already pointed out we need n to be an integer so we simply need to look at the floor and ceiling of L/e and pick the one that results in the higher value of y

another interesting approach to this problem is to look at Y as a series for fixed L and seeing which value of n gives the max

So we have for a fixed L we have

now if y(k) is a local max then we need
y(k)>y(k-1) and y(k)>y(k+1)
this gives us
(L/k)^k>=(L/(k-1))^(k-1) and (L/k)^k>=(L/(k+1))^(k+1)
simplifying the two equations gives us

So what this means is for there to be a local max at n=k then we need for L to be in the interval


So another approach to solving this is to simply compute these intervals for various values of k and seeing which interval contains your value of L and that gives you the value of n for which the maximum occurs.

  Posted by Daniel on 2018-06-05 11:08:00
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