Let $x_1 , x_2 , x_3 , x_4 , x_5$ be non-negative real numbers such that $x_1 + x_2 + x_3 + x_4 + x_5=5$ . Determine the maximum value of $x_1x_2+x_2x_3+x_3x_4+x_4x_5$.
Normally in such questions I use the fact that the equation is symmetric and thus extremum is attained when all the variables are equal , but this can not be done here , and I have spent quite a long time on this but nothing worth-mentioning came to my mind .
Could someone please help me find the maximum value ?
Thanks !
Best Answer
$x_1x_2+x_2x_3+x_3x_4+x_4x_5=(x_1+x_3+x_5)(x_2+x_4) - (x_2x_5+x_1x_4) $
Now we try to find maximum value of $(x_1+x_3+x_5)(x_2+x_4)$ when $(x_1+x_3+x_5)+(x_2+x_4)=5$ And try to minimize the value of $(x_2x_5+x_1x_4) $.
Take, $a=(x_1+x_3+x_5)$ and $b=(x_2+x_4)$ By , A.M. $\ge $ G.M. $\implies$ $\sqrt(ab) \le \frac{a+b}{2} \implies (ab) \le (\frac{5}{2})^2 $ So, max value of $(x_1+x_3+x_5)(x_2+x_4)$ is $(\frac{5}{2})^2 $
And , clearly, minimum value of $(x_2x_5+x_1x_4) $ is $0$. So , max value of $x_1x_2+x_2x_3+x_3x_4+x_4x_5$ is $(\frac{5}{2})^2 $