Let x, y, z be positive real numbers. Which o | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) For $x, y, z > 0$,we analyze each condition:$I.$ $x^3+y^3+z^3-3xyz = \frac{1}{2}(x+y+z)((x-y)^2+(y-z)^2+(z-x)^2) = 0$. Since $x+y+z > 0$,this impl

Vedclass · Accepted Answer

$I, II, IV$ only

Vedclass · Answer

(B) For $x, y, z > 0$,we analyze each condition:$I.$ $x^3+y^3+z^3-3xyz = \frac{1}{2}(x+y+z)((x-y)^2+(y-z)^2+(z-x)^2) = 0$. Since $x+y+z > 0$,this implies $(x-y)^2+(y-z)^2+(z-x)^2 = 0$,so $x=y=z$.$II.$ $x^3+y^2z+yz^2=3xyz$. Dividing by $xyz$,we get $\frac{x^2}{yz} + \frac{y}{x} + \frac{z}{x} = 3$. By $AM$-$GM$ inequality,$\frac{x^2}{yz} + \frac{y}{x} + \frac{z}{x} \ge 3 \sqrt[3]{\frac{x^2}{yz} \cdot \frac{y}{x} \cdot \frac{z}{x}} = 3(1) = 3$. Equality holds if $\frac{x^2}{yz} = \frac{y}{x} = \frac{z}{x}$,which implies $x=y=z$.$III.$ $x^3+y^2z+z^2x=3xyz$. Let $x=1, y=2, z=0.5$. Then $1 + (4)(0.5) + (0.25)(1) = 1 + 2 + 0.25 = 3.25 
eq 3(1)(2)(0.5) = 3$. However,testing values like $x=1, y=1, z=1$ gives $1+1+1=3$. This condition does not force $x=y=z$ for all positive reals.$IV.$ By $AM$-$GM$,$\frac{x+y+z}{3} \ge \sqrt[3]{xyz}$. Thus $(x+y+z)^3 \ge 27xyz$. Equality holds if and only if $x=y=z$.Thus,$I, II,$ and $IV$ imply $x=y=z$.

Let $x, y, z$ be positive real numbers. Which of the following conditions imply $x=y=z$?
$I.$ $x^3+y^3+z^3=3xyz$
$II.$ $x^3+y^2z+yz^2=3xyz$
$III.$ $x^3+y^2z+z^2x=3xyz$
$IV.$ $(x+y+z)^3=27xyz$

Similar Questions

The minimum value of $\frac{9 \cdot 3^{2x} + 6 \cdot 3^x + 4}{9 \cdot 3^{2x} - 6 \cdot 3^x + 4}$ is

The set of all real values $a$ for which $-1 < \frac{2 x^2+a x+2}{x^2+x+1} < 3$ holds for all real values of $x$ is

If the minimum value of $f(x)=2x^2+\alpha x+8$ is the same as the maximum value of $g(x)=-3x^2-4x+\alpha^2$,then $\alpha^2=$

The sum of the squares of the roots of $|x-2|^2+|x-2|-2=0$ and the squares of the roots of $x^2-2|x-3|-5=0$ is:

The sum of the squares of all the roots of the equation $x^2+|2x-3|-4=0$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module

Let $x, y, z$ be positive real numbers. Which of the following conditions imply $x=y=z$?$I.$ $x^3+y^3+z^3=3xyz$$II.$ $x^3+y^2z+yz^2=3xyz$$III.$ $x^3+y^2z+z^2x=3xyz$$IV.$ $(x+y+z)^3=27xyz$