Proving the equivalence of two statements

[Lionel]

Good evening all,

I have two statements which are supposedly equivalent; I'd like to demonstrate so.

(P): (a+b / 2)^3 <= (a^3 + b^3) / 2
(Q): ab(a+b) <= a^3 + b^3

I'm really stuck, I tried to simply as much as possible the first statement by factoring, didn't seem to work. Any help?

Best regards

 

[stapel]

I have two statements which are supposedly equivalent; I'd like to demonstrate so.

What do you mean by "demonstrating so"?

(P): (a+b / 2)^3 <= (a^3 + b^3) / 2
(Q): ab(a+b) <= a^3 + b^3

Your formatting of (P) is ambiguous. Do you mean the following?

. . . . .\(\displaystyle \mbox{P: }\, \left(\, \dfrac{a\, +\, b}{2}\, \right)^3\, \leq\, \dfrac{a^3\, +\, b^3}{2}\)

I tried to simply as much as possible the first statement by factoring, didn't seem to work.

When you reply, please include a clear listing of what you have tried so far. Thank you!