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 Algebra Pre-Algebra and Basic Algebra Math Forum

 April 11th, 2014, 09:05 AM #1 Senior Member   Joined: Nov 2013 Posts: 434 Thanks: 8 show Show that if real numbers x,y,z>0 satisfy the relation xyz+xy+yz+zx=4, then x+y+z>= 3. April 11th, 2014, 09:20 AM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,681 Thanks: 2659 Math Focus: Mainly analysis and algebra The first equation gives us $$\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 3$$ Which I think turns out to be useful in this one. April 11th, 2014, 09:34 AM #3 Global Moderator   Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms That doesn't look quite right... Thanks from v8archie April 11th, 2014, 12:22 PM   #4
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 Originally Posted by v8archie The first equation gives us $$\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 3$$ Which I think turns out to be useful in this one.
Right side is wrong. Should be 4/(xyz) - 1 April 11th, 2014, 01:27 PM #5 Senior Member   Joined: Apr 2014 From: Greater London, England, UK Posts: 320 Thanks: 156 Math Focus: Abstract algebra Let $t=x+y+z$. By AM–GM, $$t^3\ =\ (x+y+z)^3\ \geqslant\ 27xyz\ \ldots\fbox1$$ By Cauchy–Schwarz, \begin{align*} t^2\ =\ (x+y+z)^2 &\leqslant\ (x^2+y^2+z^2)(1^2+1^2+1^2) \\\\ &=\ 3(x^2+y^2+z^2) \\\\ &=\ 3t^2-6(xy+yz+zx)\end{align*} $\displaystyle \therefore\ t^2\ \geqslant\ 3(xy+yz+zx)\ =\ 12-3xyz\ \ldots\fbox2$ $\displaystyle \fbox1+9\times\fbox2$ gives \begin{align*} t^3+3t^2-108 &\geqslant\ 0 \\\\ (t-3)(t+6)^2 &\geqslant\ 0 \end{align*} Since $(t+6)^2>0$, it follows that $t-3\geqslant0$. QED Thanks from v8archie April 11th, 2014, 05:32 PM #6 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,681 Thanks: 2659 Math Focus: Mainly analysis and algebra Oops. I should answer when I have a bit of time to think more clearly! Tags show 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 Shamieh Algebra 5 July 3rd, 2013 06:35 PM the_liong Algebra 1 September 12th, 2010 12:46 AM notnaeem Real Analysis 4 August 16th, 2010 12:32 PM john235 Real Analysis 2 March 2nd, 2009 07:48 AM peka0027 Real Analysis 2 February 26th, 2009 06:01 PM

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