If `a, b, c` are distinct positive real numbers such that the quadratic expression `Q_(1)(x) = ax^(2) + bx + c`,
`Q_(2)(x) = bx^(2) + cx + a, Q_(3)(x) = cx^(2) + ax + b` are always non-negative, then possible integer in the range of the expression `y = (a^(2)+ b^(2) + c^(2))/(ab + bc + ca)` is

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to analyze the conditions given for the quadratic expressions and derive the range for the expression \( y = \frac{a^2 + b^2 + c^2}{ab + bc + ca} \).

### Step-by-Step Solution:

1. **Understanding the Quadratic Conditions**:
We have three quadratic expressions:
\[ Q_1(x) = ax^2 + bx + c, \quad Q_2(x) = bx^2 + cx + a, \quad Q_3(x) = cx^2 + ax + b \]
Each of these must be non-negative for all \( x \). This implies that their discriminants must be less than or equal to zero.

2. **Finding Discriminants**:
- For \( Q_1(x) \):
\[ D_1 = b^2 - 4ac \leq 0 \quad \text{(Equation 1)} \]
- For \( Q_2(x) \):
\[ D_2 = c^2 - 4ab \leq 0 \quad \text{(Equation 2)} \]
- For \( Q_3(x) \):
\[ D_3 = a^2 - 4bc \leq 0 \quad \text{(Equation 3)} \]

3. **Combining the Inequalities**:
From the three inequalities, we can derive:
\[ b^2 \leq 4ac \quad \text{(from Equation 1)} \]
\[ c^2 \leq 4ab \quad \text{(from Equation 2)} \]
\[ a^2 \leq 4bc \quad \text{(from Equation 3)} \]

4. **Summing the Inequalities**:
Adding all three inequalities gives:
\[ b^2 + c^2 + a^2 \leq 4(ac + ab + bc) \]
Rearranging this, we have:
\[ a^2 + b^2 + c^2 \leq 4(ab + ac + bc) \quad \text{(Equation 4)} \]

5. **Dividing by the Sum**:
Dividing both sides of Equation 4 by \( ab + ac + bc \) (which is positive since \( a, b, c \) are positive):
\[ \frac{a^2 + b^2 + c^2}{ab + ac + bc} \leq 4 \]

6. **Finding the Lower Bound**:
Now, we need to find a lower bound for \( y \):
- Using the Cauchy-Schwarz inequality:
\[ (a^2 + b^2 + c^2)(1 + 1 + 1) \geq (a + b + c)^2 \]
This implies:
\[ a^2 + b^2 + c^2 \geq \frac{(a + b + c)^2}{3} \]
Thus,
\[ \frac{a^2 + b^2 + c^2}{ab + ac + bc} \geq 1 \quad \text{(Equation 5)} \]

7. **Combining Results**:
From Equations 4 and 5, we have:
\[ 1 < \frac{a^2 + b^2 + c^2}{ab + ac + bc} \leq 4 \]
Therefore, the possible integer values for \( y \) are \( 2, 3, \) and \( 4 \).

### Conclusion:
The possible integer values in the range of the expression \( y = \frac{a^2 + b^2 + c^2}{ab + ac + bc} \) are \( 2, 3, \) and \( 4 \).