We are tasked with finding the range of the following expression involving three distinct positive real numbers $x$, $y$, and $z$:

$S = \frac{9(x^2 + y^2)}{4z^2} + \frac{9(y^2 + z^2)}{4x^2} + \frac{9(x^2 + z^2)}{4y^2}.$

Step 1: Symmetry in the expression

The expression $S$ is symmetric in terms of $x$, $y$, and $z$, so we expect the minimum value to occur when $x = y = z$, and then we can explore the general behavior.

Step 2: Substituting $x = y = z$

Let $x = y = z = t$. Substituting this into the expression for $S$:

= 3 \times \frac{9(2t^2)}{4t^2} = 3 \times \frac{18}{4} = 3 \times 4.5 = 13.5.$$ So, when $$x = y = z$$, $$S = 13.5$$. ### Step 3: Investigating the range of $$S$$ Now, let's explore the behavior when $$x$$, $$y$$, and $$z$$ are distinct positive real numbers. Since $$x^2 + y^2 \geq 2xy$$ by the arithmetic mean-geometric mean inequality (AM-GM inequality), each term in the expression $$S$$ is positive. Additionally, since the denominator involves the squares of $$x$$, $$y$$, and $$z$$, making one of the variables very small compared to the others will cause one of the terms to grow without bound. Thus, as $$x$$, $$y$$, or $$z$$ approach zero or infinity, the expression $$S$$ increases indefinitely. ### Step 4: Conclusion - The minimum value of $$S$$ occurs when $$x = y = z$$, and it is $$13.5$$. - As $$x$$, $$y$$, or $$z$$ become large or small, $$S$$ can become arbitrarily large. Therefore, the range of $$S$$ is: $$[13.5, \infty).$$ Would you like further clarifications, or have additional questions? Here are 5 related questions you might find interesting: 1. What is the AM-GM inequality, and how is it applied in optimization problems? 2. Can this technique be generalized to higher-order symmetric expressions? 3. What is the behavior of the expression if $$x$$, $$y$$, and $$z$$ are not distinct? 4. How would the range change if $$x$$, $$y$$, and $$z$$ are constrained within a specific interval? 5. Can similar methods be used to explore other types of inequalities involving symmetric functions? **Tip:** Symmetry in mathematical expressions often provides a way to simplify and minimize or maximize functions, making it easier to solve optimization problems.