Let $x,y,z$ be positive real numbers such that $x+y+z=3$. Prove that
$8(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})+9\geq 10(x^{2}+y^{2}+z^{2})$.
Which way to go?
I need the solution of this problem.
Let $x,y,z$ be positive real numbers such that $x+y+z=3$. Prove that
$8(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})+9\geq 10(x^{2}+y^{2}+z^{2})$.
Let $x,y,z$ be positive real numbers such that $x+y+z=3$. Prove that
$8(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})+9\geq 10(x^{2}+y^{2}+z^{2})$.
$\color{blue}{\textit{To}} \color{red}{\textit{ problems }} \color{blue}{\textit{I am encountering with-}} \color{green}{\textit{AVADA KEDAVRA!}}$
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sakibtanvir
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Re: Which way to go?
At first I thought the problem would require some serious theorems.But my thought got in vain.You know,I know a little in inequalities though I tried what my best to find it with simple way.
$8(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})+9 \geq 10(x^{2}+y^{2}+z^{2})$
$\rightarrow8(xy+yz+zx)+9xyz \geq 10(x^{2}+y^{2}+z^{2})xyz$
$\rightarrow8(xy+yz+zx) \geq xyz(10(x^{2}+y^{2}+z^{2})-9)$
$\rightarrow8(xy+yz+zx) \geq xyz(10(x+y+z)^{2}-20(xy+yz+zx)-9)$
$\rightarrow8(xy+yz+zx) \geq xyz(90-9-20(xy+yz+zx))$
$\rightarrow8(xy+yz+zx) \geq 81xyz-20(xy+yz+zx)xyz$
$\rightarrow8(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})+20(xy+yz+zx) \geq 81$
Can you prove it from there?
$8(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})+9 \geq 10(x^{2}+y^{2}+z^{2})$
$\rightarrow8(xy+yz+zx)+9xyz \geq 10(x^{2}+y^{2}+z^{2})xyz$
$\rightarrow8(xy+yz+zx) \geq xyz(10(x^{2}+y^{2}+z^{2})-9)$
$\rightarrow8(xy+yz+zx) \geq xyz(10(x+y+z)^{2}-20(xy+yz+zx)-9)$
$\rightarrow8(xy+yz+zx) \geq xyz(90-9-20(xy+yz+zx))$
$\rightarrow8(xy+yz+zx) \geq 81xyz-20(xy+yz+zx)xyz$
$\rightarrow8(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})+20(xy+yz+zx) \geq 81$
Can you prove it from there?
Last edited by sakibtanvir on Wed Sep 12, 2012 1:11 pm, edited 5 times in total.
An amount of certain opposition is a great help to a man.Kites rise against,not with,the wind.
Re: Which way to go?
Sorry 
but would you please check if there is some typo? 
but would you please check if there is some typo? $\color{blue}{\textit{To}} \color{red}{\textit{ problems }} \color{blue}{\textit{I am encountering with-}} \color{green}{\textit{AVADA KEDAVRA!}}$
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sakibtanvir
- Posts:188
- Joined:Mon Jan 09, 2012 6:52 pm
- Location:24.4333°N 90.7833°E
Re: Which way to go?
Sorry for the foolish typo. I hate latexing, I had to edit 4 times in total!
I hate latexing, I had to edit 4 times in total!An amount of certain opposition is a great help to a man.Kites rise against,not with,the wind.