If, s, r and t are three positive real numbers then find the minimum value of \frac{(5r^{2}+r+5)(7s^{2}+s+7)(9t^{2}+t+9)}{57rst}

We are tasked with finding the minimum value of the expression:

\[
\frac{(5r^{2} + r + 5)(7s^{2} + s + 7)(9t^{2} + t + 9)}{57rst}
\]
where \( r \), \( s \), and \( t \) are positive real numbers.

### Step 1: Analyze the structure of the terms
The given expression involves quadratic terms for each variable \( r \), \( s \), and \( t \), along with linear and constant terms. We aim to find the minimum of the entire product.

First, let's break down the form of each factor in the numerator:
- \( 5r^2 + r + 5 \)
- \( 7s^2 + s + 7 \)
- \( 9t^2 + t + 9 \)

By using an optimization method like **AM-GM inequality**, we can attempt to minimize the expression.

### Step 2: Apply AM-GM inequality to each term
For each quadratic term, let's apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality to find the minimum values.

#### For \( 5r^2 + r + 5 \):
We express it as:
\[
5r^2 + r + 5 = 5 \left( r^2 + \frac{r}{5} + 1 \right)
\]
By the AM-GM inequality:
\[
r^2 + \frac{r}{5} + 1 \geq 3 \sqrt[3]{r^2 \cdot \frac{r}{5} \cdot 1}
\]
The minimum occurs when all terms are equal. Solving this, we find the minimum value of \( 5r^2 + r + 5 \) occurs when \( r = 1 \), giving \( 5r^2 + r + 5 = 11 \).

#### For \( 7s^2 + s + 7 \):
We express it similarly:
\[
7s^2 + s + 7 = 7 \left( s^2 + \frac{s}{7} + 1 \right)
\]
Applying AM-GM inequality:
\[
s^2 + \frac{s}{7} + 1 \geq 3 \sqrt[3]{s^2 \cdot \frac{s}{7} \cdot 1}
\]
The minimum occurs when \( s = 1 \), giving \( 7s^2 + s + 7 = 15 \).

#### For \( 9t^2 + t + 9 \):
We express it as:
\[
9t^2 + t + 9 = 9 \left( t^2 + \frac{t}{9} + 1 \right)
\]
Applying AM-GM inequality:
\[
t^2 + \frac{t}{9} + 1 \geq 3 \sqrt[3]{t^2 \cdot \frac{t}{9} \cdot 1}
\]
The minimum occurs when \( t = 1 \), giving \( 9t^2 + t + 9 = 19 \).

### Step 3: Compute the minimum value
At \( r = s = t = 1 \), the expression simplifies to:

\[
\frac{(11)(15)(19)}{57 \times 1 \times 1 \times 1} = \frac{11 \times 15 \times 19}{57}
\]
First, calculate the product in the numerator:
\[
11 \times 15 = 165
\]
\[
165 \times 19 = 3135
\]
Thus, the expression becomes:
\[
\frac{3135}{57} = 55
\]

### Final Answer:
The minimum value of the given expression is \( \boxed{55} \).

Would you like further details or have any questions?

Here are 5 related questions:
1. What is the AM-GM inequality and how is it used in optimization?
2. Can the same method be used to minimize other expressions involving products of quadratics?
3. Why do \( r = s = t = 1 \) yield the minimum values in this case?
4. How would the result change if the coefficients in the quadratic terms were different?
5. What other inequalities could be used to solve similar problems?

**Tip**: The AM-GM inequality is a powerful tool for estimating minimum values when dealing with sums and products of positive numbers.

Find the minimum value of the expression involving quadratic functions using the AM-GM inequality. This step-by-step guide explains how to apply optimization techniques to minimize the expression (5r^2 + r + 5)(7s^2 + s + 7)(9t^2 + t + 9) / 57rst. Suitable for college-level students.

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