Let `a`, `b`, `c` and `m in R^(+)`. The possible value of `m` (independent of `a`, `b` and `c`) for which atleast one of the following equations have real roots is
`{:(ax^(2)+bx+cm=0),(bx^(2)+cx+am=0),(cx^(2)+ax+bm=0):}}`

Options

A. `(1)/(2)`
B. `(1)/(8)`
C. `(1)/(12)`
D. `(1)/(4)`

Explanation

To solve the problem, we need to find the possible values of \( m \) such that at least one of the following quadratic equations has real roots:

1. \( ax^2 + bx + cm = 0 \)
2. \( bx^2 + cx + am = 0 \)
3. \( cx^2 + ax + bm = 0 \)

### Step 1: Understanding the Condition for Real Roots
For a quadratic equation \( Ax^2 + Bx + C = 0 \) to have real roots, the discriminant \( D \) must be greater than or equal to zero:
\[ D = B^2 - 4AC \geq 0 \]

### Step 2: Calculate the Discriminants
Let's calculate the discriminants for each of the three equations.

1. For \( ax^2 + bx + cm = 0 \):
\[ D_1 = b^2 - 4a(cm) = b^2 - 4acm \]

2. For \( bx^2 + cx + am = 0 \):
\[ D_2 = c^2 - 4b(am) = c^2 - 4abm \]

3. For \( cx^2 + ax + bm = 0 \):
\[ D_3 = a^2 - 4c(bm) = a^2 - 4cbm \]

### Step 3: Combine the Conditions
We need at least one of these discriminants to be non-negative:
\[ D_1 \geq 0 \quad \text{or} \quad D_2 \geq 0 \quad \text{or} \quad D_3 \geq 0 \]

This leads us to the combined inequality:
\[ (b^2 - 4acm) + (c^2 - 4abm) + (a^2 - 4cbm) \geq 0 \]
Simplifying this gives:
\[ a^2 + b^2 + c^2 - 4m(ab + ac + bc) \geq 0 \]

### Step 4: Rearranging the Inequality
Rearranging the inequality, we have:
\[ a^2 + b^2 + c^2 \geq 4m(ab + ac + bc) \]
Dividing both sides by \( ab + ac + bc \) (which is positive since \( a, b, c > 0 \)):
\[ \frac{a^2 + b^2 + c^2}{ab + ac + bc} \geq 4m \]

### Step 5: Finding the Minimum Value
Using the Cauchy-Schwarz inequality, we know:
\[ (a^2 + b^2 + c^2)(1 + 1 + 1) \geq (a + b + c)^2 \]
This implies:
\[ a^2 + b^2 + c^2 \geq \frac{(a + b + c)^2}{3} \]
Thus, we can conclude:
\[ \frac{a^2 + b^2 + c^2}{ab + ac + bc} \geq 1 \]
This means:
\[ 1 \geq 4m \quad \Rightarrow \quad m \leq \frac{1}{4} \]

### Step 6: Considering the Positive Real Numbers
Since \( m \) is a positive real number, we have:
\[ 0 < m \leq \frac{1}{4} \]

### Final Answer
The possible values of \( m \) are:
\[ m \in (0, \frac{1}{4}] \]