How could i prove this?

Question

i want to know if that is true, if yes, how to mathematically prove it

if the sum of some (fixed numbers of)number is fixed, their product is biggest when all those numbers are equal,and in contrast, if the product of the numbers are determined, the sum is smallest if those numbers are equal

thanks

Answer 1

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Answer 2

As long as we restrict to positive numbers, I think you can prove it in about two lines using GM ≤ AM.  

And Brian B was already on it... heh.

Answer 3

the questions is honestly a little confusing.  It sounds like it's not true... but i can't be sure because i'm not sure i truly understand it

Answer 4

Yes, this is true, provided that all of the numbers are non-negative. These statements are direct consequences of the AM-GM inequality, which states that for non-negative a_1, a_2, ... a_k:

(a_1 + a_2 + ... + a_k)/k >= (a_1*a_2*...*a_k)^(1/k)

here comes the important part - with equality holding if and only iff all of the a_1, a_2, ... a_k are _equal_.

This means that

(a_1*a_2*...*a_k) <= [(a_1 + a_2 + ... + a_k)/k]^k

with equality holding if and only iff all the a's are equal. Hence, when not all of the a's are equal, the left side will be less than the right side (which is a constant).

Similarly, we can write

(a_1 + a_2 + ... + a_k) >= k*(a_1*a_2*...*a_k)^(1/k)

Again, equality holds only when all the a's are equal; otherwise the left side will be larger than the right side (which is a constant in the second part of the question.)