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 November 11th, 2017, 06:41 AM #1 Senior Member   Joined: Nov 2011 Posts: 250 Thanks: 3 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, 07:41 AM #2 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,968 Thanks: 1152 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 $a^2-ab$. Under what condition is this expression a minimum? Thanks from topsquark
 November 11th, 2017, 08:09 AM #3 Senior Member   Joined: Nov 2011 Posts: 250 Thanks: 3 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, 08:12 AM #4 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,968 Thanks: 1152 Math Focus: Elementary mathematics and beyond $$a^2-ab=a(a-b)=0\text{ iff }a=b$$ Form that it follows that $ab$ is maximized. You don't "need" this; it's just one particular approach.
 November 11th, 2017, 01:59 PM #5 Senior Member   Joined: Feb 2016 From: Australia Posts: 1,838 Thanks: 653 Math Focus: Yet to find out. Maximise $z = x y$ subject to, $x + y = 10$. We could add non-negativity 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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