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A natural number $N$ should be split into how many parts so as to maximize their products.

Attempt: assuming $N = x_{1}+x_{2}+x_{3}+\cdots \cdots +x_{n}$

now using AM GM inequality

$\displaystyle x_{1}+x_{2}+x_{3}+\cdots \cdots +x_{n}\geq n\left(x_{1}\cdot x_{2}\cdot x_{3}+\cdots \cdots \cdot x_{n}\right)^{\frac{1}{n}}$

and equality hold when $x_{1}=x_{2}=x_{3}=x_{n}$

could some help me how to solve it , thank in advanced

DXT
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    Do the parts have to be natural numbers? That is an important restriction. Otherwise, the maximum is achieved by taking parts as close as possible to $e$. – Jack D'Aurizio Mar 16 '17 at 02:34
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    As close as possible to $3$ in the integers, or $e$ in the reals. Note that for $n\ge 5$, $3(n-3)>n$. – Joffan Mar 16 '17 at 02:35

1 Answers1

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We can start with:

$$N=x_1+x_2+x_3+\cdots +x_n\geq n\left(x_1\cdots x_n\right)^{\frac{1}{n}}$$

so that:

$$\left(\frac{N}{n}\right)^n\geq x_1\cdots x_n$$

Equality is reached when $x_1=\dots =x_n$, and this must be the maximum. So over $\mathbb{R}$, $x_i=\frac{N}{n}$, and we need to determine $n$.

Then:

$$\frac{d}{dn}\left(\frac{N}{n}\right)^n=\left(\frac{N}{n}\right)^n\left[\ln\left(\frac{N}{n}\right)-1\right]$$

which zeroes for $\frac{N}{n}=e$.

JMP
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