November 11th, 2017, 07:41 AM  #1 
Senior Member Joined: Nov 2011 Posts: 243 Thanks: 2  Analysis question
Sum of two numbers is 10, what are they if they have the maximum multiplication? [I know the answer, but I need the way to solve it.] 
November 11th, 2017, 08:41 AM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,884 Thanks: 1088 Math Focus: Elementary mathematics and beyond 
Let's call the two numbers $a$ and $b$. Without a loss of generality, assume $a\ge b$. Now take a look at $a^2ab$. Under what condition is this expression a minimum? 
November 11th, 2017, 09:09 AM  #3 
Senior Member Joined: Nov 2011 Posts: 243 Thanks: 2 
I need a maximum. But maybe I am wrong... I think to take this expression ["a square" minus a x b] and equal it to zero. I will fully feel to thank you, for answering, first all, and for your explanation. [if you will give it to me.] Thank you... 
November 11th, 2017, 09:12 AM  #4 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,884 Thanks: 1088 Math Focus: Elementary mathematics and beyond 
$$a^2ab=a(ab)=0\text{ iff }a=b$$ Form that it follows that $ab$ is maximized. You don't "need" this; it's just one particular approach. 
November 11th, 2017, 02:59 PM  #5 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,737 Thanks: 606 Math Focus: Yet to find out. 
Maximise $z = x y$ subject to, $x + y = 10$. We could add nonnegativity constraints too. From the first constraint, $x = 10  y$ then $z = (10  y)y$. $z_y = 10  2y = 0 \implies y = 5$ Thus $x = 5$ also. See here: https://www.desmos.com/calculator/fsahli3jgy 

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