How Do You Maximize This Complex Expression With Given Constraints?

[anemone]

Here is this week's POTW:

-----
Find the maximum of $a_1+a_2+a_3+a_4-a_1a_2-a_1a_3-a_1a_4-a_2a_3-a_2a_4-a_3a_4+a_1a_2a_3+a_1a_2a_4+a_1a_3a_4+a_2a_3a_4-a_1a_2a_3a_4$

where $|a_i|\le1,\,i=1,\,2,\,3,\,4$.

-----

Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

[anemone]

No one answered last week's problem. :( You can find the proposed solution below:

Spoiler

Let $\small P=a_1+a_2+a_3+a_4-a_1a_2-a_1a_3-a_1a_4-a_2a_3-a_2a_4-a_3a_4+a_1a_2a_3+a_1a_2a_4+a_1a_3a_4+a_2a_3a_4-a_1a_2a_3a_4$

We then see that

$\small \begin{align*}P&=a_1+a_2+a_3+a_4-a_1a_2-a_1a_3-a_1a_4-a_2a_3-a_2a_4-a_3a_4+a_1a_2a_3+a_1a_2a_4+a_1a_3a_4+a_2a_3a_4-a_1a_2a_3a_4\\&=1-(1-a_1)(1-a_2)(1-a_3)(1-a_4)\end{align*}$

given $|a_i|\le1,\,i=1,\,2,\,3,\,4$.It is obvious that $P$ is then less than or equal to $1$.

Therefore, the maximum of $P$ is $1$, occurs at $a_1=a_2=a_3=a_4=1$.