Maximizing $abcd$ with given constraints - POTW #366 May 14th, 2019

[anemone]

Here is this week's POTW:

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Given $a,b,c,d$ are real numbers such that

$ab+cd=4$
$ac+bd=8$

Find the maximum value of $abcd$.

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

 

[anemone]

Congratulations to the following members for their correct solution!(Cool)

1. castor28
2. Olinguito
3. kaliprasad

Solution from castor28:

Spoiler

The quadratic equation with roots $ab$ and $cd$ is:
$$
X^2 - (ab+cd)X + (ab)(cd) = 0
$$
Since this equation has real roots and $ab+cd=4$, we must have:
$$
4(abcd) \le (ab+cd)^2 = 16
$$
Which implies $abcd\le 4$.

Using a similar argument, the relation $ac+bd=8$ gives the weaker (larger) bound $abcd\le16$.

It remains to show that the maximum ($4$) can be attained. We let $a=1, b=2$ and require $cd=2$. This gives the system of equations:
\begin{align*}
c+2d&=8\\
cd&=2
\end{align*}
with real solutions $c=4\pm2\sqrt3,d=2\mp\sqrt3$.

The maximum value of $abcd$ is therefore $\mathbf{4}$.