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mathmath333
 one year ago
The question
mathmath333
 one year ago
The question

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ybarrap
 one year ago
Best ResponseYou've already chosen the best response.2There are are 2 1/x terms in the second product, that should be 1/z right?

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1i didnt understand ur statement

ybarrap
 one year ago
Best ResponseYou've already chosen the best response.2$$ \large \color{black}{\begin{align} (x+y+z)\left(\dfrac1x+\dfrac1y+\dfrac1{\color{red}{x}}\right)\hspace{.33em}\\~\\ \end{align}} $$ should be $$ \large \color{black}{\begin{align} (x+y+z)\left(\dfrac1x+\dfrac1y+\dfrac1z\right)\hspace{.33em}\\~\\ \end{align}} $$ right?

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1oh yes , u r correct

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1find the minimum value of \(\large \color{black}{\begin{align} (x+y+z)\left(\dfrac1x+\dfrac1y+\dfrac1z\right)\hspace{.33em}\\~\\ \end{align}}\) if \(\large \color{black}{\begin{align} \{x,y,z\}\in \mathbb{R^{>0}}\hspace{.33em}\\~\\ \end{align}}\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0why did you block me???

ybarrap
 one year ago
Best ResponseYou've already chosen the best response.2Multiplying everything out we get $$ x/y+y/x+x/z+z/x+z/y+y/z+3 $$ Using method of lagrange multipliers with objective function $$ f(x,y,z)=x/y+y/x+x/z+z/x+z/y+y/z+3 $$ and constraint $$ xyz=0 $$ We get $$ x/y=1\\ x/z=1\\ y/z=1\\ $$ So minimum is $$ x/y+y/x+x/z+z/x+z/y+y/z+3=6\times 1+3=9 $$ Are you familiar with this process? https://en.wikipedia.org/wiki/Lagrange_multiplier Note, although x>0,y>0,z>0, I used xyz=0 for the constraint since this is the lower bound of their domain. Here is the setup $$ \Lambda(x,y,z)=f(x,y,z)+\lambda(xyz0)\\ $$ The first differential is $$ \Lambda_x=1/yy/x^2+1/zz/x^2+\lambda yz=0 $$ The other differentials are similar Using the constraint and this equation I get x/y=1 The other results follow in a similar manner.

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1well actually idk calculus

misty1212
 one year ago
Best ResponseYou've already chosen the best response.1HI!! what class is this?

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1this is type of question of SAT

misty1212
 one year ago
Best ResponseYou've already chosen the best response.1just asking because i think we can precede just by thinking, although i am not sure your math teacher will like it q

misty1212
 one year ago
Best ResponseYou've already chosen the best response.1\[(x+y+z))(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})\] is completely symmetric in \(x,y,z\) by which i mean if you permute them you get the same thing another way of saying you can't tell the difference between \(x,y\) and \(z\)

misty1212
 one year ago
Best ResponseYou've already chosen the best response.1and they are all positive numbers, not negatives allowed

misty1212
 one year ago
Best ResponseYou've already chosen the best response.1so since you can't tell the difference between the numbers, it will have a minimum when they are all equal

misty1212
 one year ago
Best ResponseYou've already chosen the best response.1if \(x>1\) then \(\frac{1}{x}<1\) to balance it out, make them all 1 and you get \[(1+1+1)(1+1+1)=9\]

misty1212
 one year ago
Best ResponseYou've already chosen the best response.1like i said, it is just thinking, your math teacher might have a more complicated explanation

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2I'm not so sure about that symmetry argument but we can use AMGM inequality

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.2\[\begin{align} (x+y+z)\left(\dfrac1x+\dfrac1y+\dfrac1z\right) &= \frac{x^2+y^2}{xy}+\frac{y^2+z^2}{yz}+\frac{z^2+x^2}{zx}+3 \\~\\ &\ge \frac{2xy}{xy}+\frac{2yz}{yz}+\frac{2zx}{zx}+3\\~\\ &\ge 2+2+2+3\\~\\ \end{align}\]

misty1212
 one year ago
Best ResponseYou've already chosen the best response.1i am not either, but not only is it symmetric in x, y, z, it is also symmetric in 1/x,1/y,1/z so how could it be anything other than x = y = z = 1?
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