If the equation `x^(4)+px^(3)+qx^(2)+rx+5=0` has four positive real roots, find the maximum value of `pr`.

Explanation

To solve the problem, we need to find the maximum value of the product \( pr \) given that the polynomial \( x^4 + px^3 + qx^2 + rx + 5 = 0 \) has four positive real roots. Let's denote the roots of the polynomial as \( a, b, c, d \).

### Step 1: Use Vieta's Formulas
According to Vieta's formulas, for a polynomial of the form \( x^4 + px^3 + qx^2 + rx + 5 = 0 \):
- The sum of the roots \( a + b + c + d = -p \)
- The sum of the products of the roots taken two at a time \( ab + ac + ad + bc + bd + cd = q \)
- The sum of the products of the roots taken three at a time \( abc + abd + acd + bcd = -r \)
- The product of the roots \( abcd = 5 \)

### Step 2: Apply the AM-GM Inequality
To find the maximum value of \( pr \), we can use the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

1. **For the sum of the roots**:
\[ \frac{a + b + c + d}{4} \geq \sqrt[4]{abcd} = \sqrt[4]{5} \]
Therefore,
\[ a + b + c + d \geq 4 \sqrt[4]{5} \]
This implies:
\[ -p \geq 4 \sqrt[4]{5} \quad \Rightarrow \quad p \leq -4 \sqrt[4]{5} \]

2. **For the sum of the products of the roots taken three at a time**:
\[ \frac{abc + abd + acd + bcd}{4} \geq \sqrt[4]{(abcd)^3} = \sqrt[4]{5^3} = 5^{3/4} \]
Thus,
\[ -r \geq 5^{3/4} \quad \Rightarrow \quad r \leq -5^{3/4} \]

### Step 3: Find the Maximum Value of \( pr \)
Now we want to maximize \( pr \):
\[ pr = (-p)(-r) \geq (4 \sqrt[4]{5})(5^{3/4}) = 4 \cdot 5^{1} = 20 \]

### Step 4: Conclusion
Thus, the maximum value of \( pr \) is:
\[ \boxed{20} \]