Intuitively the product has a maximum value when the 5 parts have equal lengths. The maximum product would be the fifth of k to the power of 5.

Dear Sanjay,

I am not sure in what sense you divide $K$, but instead of 5, take $n$ parts. If you mean $x_1+ \cdots + x_n=K>0$, and all addends are positive then their product is maximal for equal $x_j=K/n, j=1,\cdots, n$. E.g. for $n=5$, we have

\[ abcde = abcd(K-a-b-c-d) \leq A^4(K-A) =: f(A),\]

where $A=A(a,b,c,d)$ is the Arithmetic mean of $a,b,c,d$. The function $f(A)$ attains the maximal value for $A=K/5$, that is $e=K/5$. Since $abcde$ is invariant with respect to permutations, by the same lines we have $a=b=c=d=e=K/5$.

Regards,

Tibor

A positive value 'K' is divided into 5 parts. So each part is equal to 'K/5'.

And their product will be equal to (K/5) raised to the power of 5 or

((K raised to 5)/3125).

Maximum and minimum values will depend on the value of 'K'.

This is an instance of the arithmetic-geometric inequality.

The arithmetic mean of positive numbers is no smaller than the geometric mean, with equality if and only if the numbers are all the same.

In this case the arithmetic mean of the 5 numbers is given and fixed as K/5, so the geometric mean is at most K/5 and this maximum is achieved only when all 5 numbers are equal to each other and to K/5.

More generally, look at Jensen's inequality and Muirhead's inequality.

You can use Lagrange multipliers to show that the product has a maximum value when the 5 parts have equal lengths. Symmetry considerations will lead to the same result. Then the maximum product is k^5/5^5.

One can state that

k = a0 k + a1 k + a2 k + a3 k + a4 k,

and therefore

1 = a0 + a1 + a2 + a3 + a4.

Then you can solve the following optimization problem (using Mathematica in this case):

NMaximize[{a0 *a1 *a2 *a3 *a4 , a0 + a1 + a2 + a3 + a4 - 1 == 0,

0 < a0 < 1, 0 < a1 < 1, 0 < a2 < 1, 0 < a3 < 1, 0 < a4 < 1}, {a0,

a1, a2, a3, a4}]

Then you get as result:

{0.00032, {a0 -> 0.2, a1 -> 0.2, a2 -> 0.2, a3 -> 0.2, a4 -> 0.2}}

Being 0.00032 equal to (1/5)^5. Conclusion: Abdallah's intuiton was right :)

I presume you are mathematicians. This fact does'n imply you use necessarily Mathematica (I must confess I am a user, when it needs). As a mathematician I remark that the problem is symmetric w.r.t. the independent variables Therefore the answer cannot be different from K/n.

Be happy (without Mathematica).

Nice answer, Lorenzo. I'm an engineer on vacation. I confess I chose Mathematica instead of a book :)

Dear Franco,

you are an engineer (although in vacation), while I am an accountant and later a mathematician. Nobody is perfect.

I'd like to talk with you about the trade-off between using Mathematica and approaching a problem as a mathematician. I like a lot Mathematica: it's like a good restaurant. But Mathematis is something else. My answer comes from Mathematics, which is inside our brainso and that non necessarily has been already censused .. in the wondeful world of Mathematica.

All the best and CIAO

Lorenzo

Dear all,

I think that this question is overdiscussed. Everyone agrees that the quantities have to be equal when their sum is prescribed, and we maximize their product. Indeed, when only two addends change their values from $x$ to $x+h, x-h, h \in (0,x)$, then beacuse

\[ x^5 - x^3(x^2-h^2) = x^3h^2>0\]

the maximal value we achieve for equal addends exclusively, etc.

Regards,

Tibor

Tibor is right. Stop!