User Name Remember Me? Password

 Pre-Calculus Pre-Calculus Math Forum

 March 2nd, 2019, 11:05 AM #1 Senior Member   Joined: Dec 2015 From: somewhere Posts: 605 Thanks: 88 Inequality proof $\displaystyle a^2 +b^2 +c^2 +d^2 =1 \; \;$ , where $\displaystyle a,b,c,d >0$ . Prove that $\displaystyle a+b+c+d -1 \geq 16abcd$ . March 4th, 2019, 02:10 PM #2 Member   Joined: Feb 2019 From: United Kingdom Posts: 44 Thanks: 3 I don't know if this helps but the sum of the squares of the sides of any parallelogram is equal to the sum of the squares of the diagonals. In your case, a^2 + b^2 + c^2 + d^2 is the sum of the squares of the sides of any parallelogram and the sum of the squares of the diagonals is equal to 1. The condition a,b,c, and d must satisfy makes "any" parallelogram not true but there is one that meets the criteria for a,b positive Let a=c and b=d, then a^2 + b^2 = 1/2 All my results are a bit meaningless and pointless really. Nothing worth showing. Thanks from idontknow March 4th, 2019, 02:39 PM #3 Senior Member   Joined: Dec 2015 From: somewhere Posts: 605 Thanks: 88 I almost proved it in a weak way : $\displaystyle 16abcd\leq 1 =a^2 +b^2 +c^2 +d^2 \leq a+b+c+d$ . But the inequality contains a -1 to the right side . March 4th, 2019, 03:02 PM #4 Member   Joined: Oct 2018 From: USA Posts: 88 Thanks: 61 Math Focus: Algebraic Geometry I've been working at this for a while now and that (-1) has messed me up every time. I do know that $a+b+c+d \in (1,2]$ and $16abcd \in (0,1]$, but I can't seem to link them together to account for the (-1). Since the constraint is a sphere maybe there would be a way to do some weird trigonometry on it? Thanks from idontknow Last edited by skipjack; March 4th, 2019 at 04:06 PM. March 4th, 2019, 03:04 PM   #5
Member

Joined: Feb 2019
From: United Kingdom

Posts: 44
Thanks: 3

Quote:
 Originally Posted by idontknow I almost proved it in a weak way : $\displaystyle 16abcd\leq 1 =a^2 +b^2 +c^2 +d^2 \leq a+b+c+d$ . But the inequality contains a -1 to the right side .
No way.

Let's say you don't know what they're asking you to show. You must derive it. March 4th, 2019, 03:25 PM #6 Senior Member   Joined: Dec 2015 From: somewhere Posts: 605 Thanks: 88 Yes the constraint is a sphere and trigonometry may be useful . Another way is to use Lagrange-multipliers . Tags inequality, proof 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 idontknow Elementary Math 2 November 5th, 2017 01:16 AM klim Real Analysis 0 December 20th, 2014 05:22 AM KyVanchhay Math Events 0 July 27th, 2013 01:24 AM eChung00 Applied Math 10 March 6th, 2013 10:28 AM chaolun Real Analysis 2 April 22nd, 2011 12:10 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top

Copyright © 2019 My Math Forum. All rights reserved.       