If x,y,z are positive and x+y+z =6 what is the range of value of (x/y)+ (y/z)+z(/x)

Given that \( x \), \( y \), and \( z \) are positive numbers and \( x + y + z = 6 \), we are asked to find the range of the expression:

\[
\frac{x}{y} + \frac{y}{z} + \frac{z}{x}
\]

To approach this, we can apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality, which states that for any positive real numbers \( a \), \( b \), and \( c \):

\[
\frac{a}{b} + \frac{b}{c} + \frac{c}{a} \geq 3
\]

Equality holds in the AM-GM inequality if and only if \( \frac{x}{y} = \frac{y}{z} = \frac{z}{x} \), implying \( x = y = z \).

### Step-by-Step Solution:

1. **Apply AM-GM Inequality**:
   Using AM-GM inequality directly on our expression:

   \[
   \frac{x}{y} + \frac{y}{z} + \frac{z}{x} \geq 3
   \]

   This gives us a lower bound of 3.

2. **Evaluate When \( x = y = z \)**:
   If \( x = y = z \), then from \( x + y + z = 6 \) we get \( 3x = 6 \) or \( x = 2 \). Thus, \( x = y = z = 2 \).

   Substituting \( x = y = z = 2 \) into the expression:

   \[
   \frac{x}{y} + \frac{y}{z} + \frac{z}{x} = \frac{2}{2} + \frac{2}{2} + \frac{2}{2} = 1 + 1 + 1 = 3
   \]

   So, the minimum value of the expression is 3.

3. **Explore the Upper Bound**:
   For the upper bound, we consider that as \( x \), \( y \), and \( z \) vary under the constraint \( x + y + z = 6 \), the expression \( \frac{x}{y} + \frac{y}{z} + \frac{z}{x} \) can increase significantly if there is a large disparity among \( x \), \( y \), and \( z \).

   However, without specific values, the upper bound is challenging to determine precisely. In many cases, with positive numbers and constraints like \( x + y + z = 6 \), this expression does not have a finite upper bound as the terms can diverge if one variable approaches zero. Thus, the expression can grow arbitrarily large.

### Conclusion

The range of the expression \( \frac{x}{y} + \frac{y}{z} + \frac{z}{x} \) is:

\[
[3, \infty)
\]

Would you like further details on any specific steps? 

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### Related Questions
1. How would the range change if \( x \), \( y \), and \( z \) were bounded differently?
2. What happens if we add additional constraints to \( x \), \( y \), and \( z \)?
3. How can the AM-GM inequality be applied to other expressions with similar forms?
4. How would we approach this if \( x \), \( y \), and \( z \) were restricted to integer values?
5. Can we use calculus to explore the range for a minimum or maximum value?

**Tip**: AM-GM inequality is a powerful tool for establishing bounds, particularly with symmetric expressions in positive variables.

If x, y, z are positive and x + y + z = 6, what is the range of the value of (x/y) + (y/z) + (z/x)?

Explore the solution to finding the range of the expression (x/y) + (y/z) + (z/x) under the condition x + y + z = 6. This problem uses the AM-GM inequality, showing that the range is [3, ∞) with a step-by-step application of inequality principles. Ideal for advanced algebra students in Grades 11-12.

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