The function k`f(x)=a x^3+b x^2+c x+d` has three positive roots. If the sum of the roots of `f(x)` is 4, the larget possible inegal values of `c//a` is ____________.

Explanation

To solve the problem, we need to analyze the polynomial function \( f(x) = ax^3 + bx^2 + cx + d \) which has three positive roots. We are given that the sum of the roots is 4, and we need to find the largest possible integral value of \( \frac{c}{a} \).

### Step-by-Step Solution:

1. **Identify the Roots**:
Let the roots of the polynomial be \( p, q, r \). According to Vieta's formulas, the sum of the roots is given by:
\[ p + q + r = -\frac{b}{a} \]
We know from the problem statement that:
\[ p + q + r = 4 \]

2. **Express \( c \) in terms of the roots**:
From Vieta's relations, we also know:
\[ pq + qr + rp = \frac{c}{a} \]
We need to maximize \( \frac{c}{a} \).

3. **Use the Cauchy-Schwarz Inequality**:
We can apply the Cauchy-Schwarz inequality to relate the sums:
\[ (p + q + r)^2 \leq 3(pq + qr + rp) \]
Substituting \( p + q + r = 4 \):
\[ 4^2 \leq 3(pq + qr + rp) \]
This simplifies to:
\[ 16 \leq 3(pq + qr + rp) \]
Therefore:
\[ pq + qr + rp \geq \frac{16}{3} \]

4. **Finding the Maximum Value**:
Since we need \( \frac{c}{a} = pq + qr + rp \), we have:
\[ \frac{c}{a} \geq \frac{16}{3} \]
However, we want to find the largest integral value of \( \frac{c}{a} \). The largest integer less than or equal to \( \frac{16}{3} \) is 5.

5. **Conclusion**:
Thus, the largest possible integral value of \( \frac{c}{a} \) is:
\[ \boxed{5} \]