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 January 21st, 2019, 06:26 PM #1 Newbie   Joined: Jun 2014 From: Hong Kong Posts: 7 Thanks: 0 Fastest structured way to get max(abc) if a+b+c=30 What is the fastest and structured way to get maximum of abc if a+b+c=n, say n=30? a,b,c are non-negative and can be non-integer. Last edited by kelsiu; January 21st, 2019 at 06:29 PM. January 22nd, 2019, 01:02 AM #2 Global Moderator   Joined: Dec 2006 Posts: 20,942 Thanks: 2210 Choose a = b = c. Thanks from idontknow January 22nd, 2019, 05:57 AM #3 Senior Member   Joined: Dec 2015 From: somewhere Posts: 606 Thanks: 88 What about for a,b,c natural numbers ? And how to find the minimal value? I searched a bit for Lagrange-Multipliers but how to use it for natural numbers? For this type of problem AM-GM works . $\displaystyle (a+b+c)^3 \geq 3^3 abc \;\;$ , $\displaystyle abc\leq 10^3 =max(abc)$ Last edited by idontknow; January 22nd, 2019 at 06:11 AM. January 22nd, 2019, 06:08 AM #4 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,675 Thanks: 2655 Math Focus: Mainly analysis and algebra Find the minimum in the reals and test the nearest natural solutions. The minimum product should be fairly obvious. Thanks from idontknow January 22nd, 2019, 12:39 PM #5 Senior Member   Joined: Dec 2015 From: somewhere Posts: 606 Thanks: 88 Other expressions are: $\displaystyle max(abc)=\lfloor (\frac{n}{3})^3 \rfloor$ . Last edited by idontknow; January 22nd, 2019 at 12:45 PM. January 22nd, 2019, 02:02 PM #6 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038 10 + 10 + 10 = 30 10 * 10 * 10 = 1000 9.99 + 10 + 10.01 = 30 9.99 * 10 * 10.01 = 999.99899.... Get my drift? January 23rd, 2019, 10:18 PM   #7
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Quote:
 Originally Posted by kelsiu What is the fastest and structured way to get maximum of abc if a+b+c=n, say n=30? a,b,c are non-negative and can be non-integer.
Solve

$\nabla \left(a b c - \lambda(a + b + c - 30)\right)$

$b c - \lambda = 0$
$a c - \lambda = 0$
$a b - \lambda = 0$
$a+b+c = 30$

$\dfrac b a = 1$

$\dfrac c a = 1$

$\dfrac c b = 1$

$a = b = c$ (As Skipjack mentioned)

$3a = 30$
$a=b=c = 10$ January 24th, 2019, 01:25 AM #8 Global Moderator   Joined: Dec 2006 Posts: 20,942 Thanks: 2210 If p and q are any two of the variables in the original problem, the solution must maximize pq = ((p + q)² - (p - q)²)/4, where p + q is a constant, so p = q. Thanks from greg1313 January 24th, 2019, 07:07 AM #9 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126 F(x,y,z)=xyz, G(x,y,z)=x+y+z=30 dF=Fxdx+Fydy+Fzdz=0, but dx, dy, dz not arbitrary. They are subject to dG=dx+dy+dz=0 -> (Fx-Fz)dx=0, (Fy-Fz)dy=0, dx and dy arbitrary.-> Fx=Fy -> yz=xz -> x=y But I could also have let dx and dz be arbitrary. Repeating the above I would get x=z. So x=y=z=10. Last edited by skipjack; January 24th, 2019 at 11:42 AM. January 24th, 2019, 07:24 AM #10 Senior Member   Joined: Dec 2015 From: somewhere Posts: 606 Thanks: 88 How to find the minimal value (a fast way)? Tags c30, fastest, maxabc, structured Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Yash Malik Algebra 6 September 9th, 2015 12:27 AM Yash Malik Calculus 2 September 6th, 2015 04:59 AM bobby2 Math Software 8 February 4th, 2013 12:22 AM zelmac Math Books 1 April 20th, 2012 06:56 AM

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