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November 11th, 2017, 07:41 AM   #1
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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.]
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November 11th, 2017, 08:41 AM   #2
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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?
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November 11th, 2017, 09:09 AM   #3
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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...
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November 11th, 2017, 09:12 AM   #4
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$$a^2-ab=a(a-b)=0\text{ iff }a=b$$

Form that it follows that $ab$ is maximized.

You don't "need" this; it's just one particular approach.
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November 11th, 2017, 02:59 PM   #5
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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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