My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum


Thanks Tree4Thanks
Reply
 
LinkBack Thread Tools Display Modes
February 5th, 2019, 11:38 AM   #1
Senior Member
 
romsek's Avatar
 
Joined: Sep 2015
From: USA

Posts: 2,314
Thanks: 1230

Not exactly calculus but it is a sort of minimization problem

$\text{given }a,b,c >0 \text{ show that}$

$\dfrac{a^3}{b^2(5a+2b)}+\dfrac{b^3}{c^2(5b+2c)}+ \dfrac{c^3}{a^2(5c+2a)} \geq \dfrac 3 7$

The equality occurs when $a=b=c$ but how to show this is a minima?
Thanks from topsquark and idontknow
romsek is offline  
 
February 5th, 2019, 03:17 PM   #2
Senior Member
 
Joined: Dec 2015
From: iPhone

Posts: 387
Thanks: 61

By simple math , it is seen that the fractions have the same minimal value .
One of them is $\displaystyle F(a,b)=\frac{a^3 }{b^2 (5a+2b)}$ . (by calculus find the values )
To make it easier substitute for $\displaystyle x=\frac{a}{b}\; \; $ then $\displaystyle F(a,b)=\frac{x^3 }{5x+2 } \; \; , x>0$
But I think this is not a method , maybe someone else can assist.

Last edited by idontknow; February 5th, 2019 at 03:39 PM.
idontknow is offline  
February 5th, 2019, 06:12 PM   #3
Senior Member
 
mrtwhs's Avatar
 
Joined: Feb 2010

Posts: 702
Thanks: 137

Google something called the Purkiss Principle.
Thanks from idontknow
mrtwhs is offline  
February 7th, 2019, 11:33 AM   #4
Senior Member
 
Joined: Dec 2015
From: iPhone

Posts: 387
Thanks: 61

Then how to prove the equality ?
idontknow is offline  
February 7th, 2019, 12:47 PM   #5
Senior Member
 
romsek's Avatar
 
Joined: Sep 2015
From: USA

Posts: 2,314
Thanks: 1230

Quote:
Originally Posted by idontknow View Post
Then how to prove the equality ?
equality is trivial, set a=b=c
romsek is offline  
February 7th, 2019, 05:49 PM   #6
Global Moderator
 
Joined: Dec 2006

Posts: 20,298
Thanks: 1971

If $x = a/b$ and $y = b/c$, the expression is a function of $x$ and $y$ for which the minimum (for positive $x$ and $y$) can be found to occur when $x = y = 1$ with the help of calculus (partial differentiation). This should work, but might be a bit tedious to do.
skipjack is offline  
February 7th, 2019, 05:54 PM   #7
SDK
Senior Member
 
Joined: Sep 2016
From: USA

Posts: 559
Thanks: 324

Math Focus: Dynamical systems, analytic function theory, numerics
One way to rigorously show that it is a minimizer is to check whether the Hessian matrix is positive definite. While this is guaranteed to succeed, its not very enlightening and is almost certainly not the "best" solution.

Problems with this sort of symmetry are almost always some clever trick using Cauchy-Schwartz or AM-GM inequality. The change of variables proposed by Idontknow appled to all 3 fractions is probably a step in the right direction.
SDK is online now  
February 8th, 2019, 04:39 AM   #8
Senior Member
 
Joined: Dec 2015
From: iPhone

Posts: 387
Thanks: 61

No need to know the values of a,b,c or x,y,z .
$\displaystyle F=\frac{x^3}{5x+2} + \frac{y^3 }{5y+2 } +\frac{z^3}{5z+2 }$ .
And 3/7 can be the minimal , if not then it is lower then minimal , so it means finding the minimal value proves the equality .
Since a constraint is given , it takes time to use calculus .
So the minimal occurs at the same value for the three fractions , x=y=z .
Now there are many ways to get $\displaystyle a=b=c$ from $\displaystyle a/b=b/c=c/a \; $ for a,b,c > 0 .

Last edited by idontknow; February 8th, 2019 at 04:45 AM.
idontknow is offline  
February 8th, 2019, 05:28 AM   #9
Global Moderator
 
Joined: Dec 2006

Posts: 20,298
Thanks: 1971

If you don't obtain any specific values or use calculus, how would you know whether you've found a minimum, as distinct from a maximum?
Thanks from idontknow
skipjack is offline  
February 8th, 2019, 08:14 AM   #10
Senior Member
 
Joined: Dec 2015
From: iPhone

Posts: 387
Thanks: 61

The minimum occurs in a point $\displaystyle t=x=y=z$ .
a=bt ; b=ct ; c=at ; and by the sum $\displaystyle a+b+c =t(a+b+c) \; \;$ or $\displaystyle t=1$ .
idontknow is offline  
Reply

  My Math Forum > College Math Forum > Calculus

Tags
calculus, minimization, problem, sort



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Question: Percentage word problem..sort of extremistpullup Elementary Math 6 July 9th, 2011 10:56 PM
Minimization problem 2 Fantini Calculus 2 October 2nd, 2010 09:04 AM
Minimization problem Fantini Calculus 4 September 30th, 2010 05:42 PM
Geometric Minimization problem @nthony Algebra 4 May 9th, 2009 04:24 PM
Minimization problem nsa_tanay Applied Math 1 November 24th, 2008 12:25 PM





Copyright © 2019 My Math Forum. All rights reserved.