The number of triples (x, y, z) of real numbe | vedclass

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

Question

(B) Given the equation $x^4+y^4+z^4+1=4xyz$.By the Arithmetic Mean-Geometric Mean $(AM \geq GM)$ inequality for four positive numbers $x^4, y^4, z^4,

Vedclass · Accepted Answer

$4$

Vedclass · Answer

(B) Given the equation $x^4+y^4+z^4+1=4xyz$.By the Arithmetic Mean-Geometric Mean $(AM \geq GM)$ inequality for four positive numbers $x^4, y^4, z^4, 1$:$\frac{x^4+y^4+z^4+1}{4} \geq \sqrt[4]{x^4 y^4 z^4 \cdot 1}$Substituting the given equation $x^4+y^4+z^4+1 = 4xyz$:$\frac{4xyz}{4} \geq |xyz|$$xyz \geq |xyz|$This inequality holds if and only if $xyz \geq 0$. For equality to hold in $AM \geq GM$,all terms must be equal:$x^4 = y^4 = z^4 = 1$This implies $|x| = 1, |y| = 1, |z| = 1$. Thus,$x, y, z \in \{1, -1\}$.Substituting these into the original equation $x^4+y^4+z^4+1=4xyz$:$1+1+1+1 = 4xyz$ $\Rightarrow 4 = 4xyz$ $\Rightarrow xyz = 1$.The possible triples $(x, y, z)$ such that their product is $1$ are:$(1, 1, 1), (1, -1, -1), (-1, 1, -1), (-1, -1, 1)$.There are $4$ such triples.

The number of triples $(x, y, z)$ of real numbers satisfying the equation $x^4+y^4+z^4+1=4xyz$ is

Similar Questions

Let $\lambda \in R$ and let the equation $E$ be $|x|^2 - 2|x| + |\lambda - 3| = 0$. Then the largest element in the set $S = \{x + \lambda : x \text{ is an integer solution of } E\}$ is $..........$

Let $m$ and $n$ be the numbers of real roots of the quadratic equations $x^2-12x+[x]+31=0$ and $x^2-5|x+2|-4=0$ respectively,where $[x]$ denotes the greatest integer $\leq x$. Then $m^2+mn+n^2$ is equal to $..............$.

If $a, b, c \in \mathbb{R}$ and $1$ is a root of the equation $ax^2 + bx + c = 0$,then the curve $y = 4ax^2 + 3bx + 2c$ $(a \neq 0)$ intersects the $x$-axis at

If $p$ and $q$ are positive real numbers such that $p^{2} + q^{2} = 1$,then the maximum value of $(p+q)$ is:

If $x$ is a real number,what are the maximum and minimum values of the expression $\frac{x^2 - 3x + 4}{x^2 + 3x + 4}$?

Vedclass Test Series

Exam Paper Generator

Online Exam Module