a,b,c>0 prove the equation have 1 root :

d^3 - d(ab+bc+ca) - 2abc =0 .

please help me , thanks :D

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- Oct 15th 2005, 03:30 PMmathhelp me
a,b,c>0 prove the equation have 1 root :

d^3 - d(ab+bc+ca) - 2abc =0 .

please help me , thanks :D - Oct 15th 2005, 10:47 PMrgep
If a cubic f has three real roots then it has a (real) maximum and a minimum between the roots. So the derivative f' has to have two real roots, and the value of f at the maximum must be positive and the value at the minimum must be negative. In this case f' is 3d^2 - (ab+bc+ca). At d = sqrt((ab+bc+ca)/3) we have f = d(d^2 - (ab+bc+ca)) - 2abc = -(2/3)(ab+bc+ba)^(3/2) - 2abc and this is indeed negative. At d = -sqrt((ab+bc+ca)/2) we have f = d(d^2 - (ab+bc+ca)) - 2abc = (2/3)(ab+bc+ba)^(3/2) - 2abc. Now (ab+bc+bc)/3 >= (ab.bc.ca)^(1/3) by the AM-GM inequality and this shows that the value at the maximum is negative unless there is equality in the AM-GM which happens only when a,b,c are all equal. We conclude that if a=b=c there are three real roots, at -a,-a and 2a; if a,b,c are not all equal there is one real root which lies to the right of sqrt((ab+bc+ca)/3).

- Oct 15th 2005, 10:58 PMmath
thansk to rgeq you are very wonderful , and second question

from the equation and x,y,z >0 . ax+by+cz=xyz . Prove the inequality

x+y+z >= 2/d . (\sqrt[(a+d)(b+d)(c+d)]) - Oct 17th 2005, 01:59 AMmathQuote:

Originally Posted by**rgep**

the root of equation ???? i don't understand why we get d = sqrt((ab+bc+ca)/3) and d = -sqrt((ab+bc+ca)/2) - Oct 17th 2005, 12:09 PMrgep
Typo: it's + or - sqrt((ab+bc+ca)/

**3**) - Oct 20th 2005, 08:23 PMmathQuote:

Originally Posted by**math**

- Oct 20th 2005, 08:27 PMmath
x,y,z >0 . ax+by+cz=xyz . with 2/d = 1/(a+d)+1/(b+d)+1/(c+d)Prove the inequality

x+y+z >= 2/d . (\sqrt[(a+d)(b+d)(c+d)]) :D :D :D - Oct 26th 2005, 02:44 PMmath
:( :( :( :(

next week i will give the proof to my teacher !!! please - Oct 26th 2005, 03:01 PMMathGuru
Maybe you should make a new thread for your second question because people usually try to answer posts with 0 replies.

- Oct 27th 2005, 05:16 AMmath
all member can't help me !!! hix hix hix

- Oct 27th 2005, 06:29 AMMathGuru
Seriously,

Start a new thread and more people will see it. Or maybe private message rgep for some more help. Few will be reading this because it is an old thread with many replies. - Oct 27th 2005, 02:40 PMmath
okie i know .

put 2/d = 1/a+d + 1/b+d + 1/c+d

and x,y,z>0 such that : ax+by+cz=xyz . Prove the inequality

(x+y+z)>=\sqrt{(a+d).(b+d).(c+d)} . 2/d

:o - Oct 27th 2005, 03:53 PMJamesonQuote:

Originally Posted by**math**

You didn't have parantheses, so I'll have to assume what you meant.

Put $\displaystyle \frac{2}{d}=\frac{1}{a+d}+\frac{1}{b+d}+\frac{1}{c +d}$

and $\displaystyle x,y,z>0$ such that: $\displaystyle ax+by+cz=xyz$.

Prove the inequality $\displaystyle x+y+z\ge \frac{2\sqrt{(a+d)(b+d)(c+d)}}{d}$

Is that what you meant? - Oct 27th 2005, 08:03 PMmath
yes , that's right . do you have proof ????

- Jan 15th 2006, 01:11 AMkonlsy
it is difficult. but i will try to sovel it. i hope i can sovel this puzzl ;)