If `ax^(2)+bx+c=0` have two distinct roots lying int eh interval `(0,1),a,b,c ` ` epsilon N` The least value of `b` is

Options

A. 5
B. 6
C. 7
D. 8

Explanation

To solve the problem, we need to find the least value of \( b \) for the quadratic equation \( ax^2 + bx + c = 0 \) such that it has two distinct roots lying in the interval \( (0, 1) \), where \( a, b, c \in \mathbb{N} \).

### Step 1: Conditions for Roots
For the quadratic equation to have two distinct roots in the interval \( (0, 1) \), the following conditions must be satisfied:
1. \( f(0) > 0 \)
2. \( f(1) > 0 \)
3. The discriminant \( D = b^2 - 4ac > 0 \)

### Step 2: Evaluate \( f(0) \) and \( f(1) \)
- \( f(0) = c > 0 \) (since \( c \in \mathbb{N} \))
- \( f(1) = a + b + c > 0 \)

Since \( a, b, c \) are natural numbers, both conditions are satisfied.

### Step 3: Expressing the Roots
Let the roots of the equation be \( \alpha \) and \( \beta \). Since both roots lie in the interval \( (0, 1) \), we can express the roots as:
- \( \alpha + \beta = -\frac{b}{a} \)
- \( \alpha \beta = \frac{c}{a} \)

### Step 4: Conditions from the Roots
From the conditions of the roots lying in \( (0, 1) \):
- \( 0 < \alpha < 1 \)
- \( 0 < \beta < 1 \)

This implies:
- \( 0 < \alpha + \beta < 2 \)
- \( 0 < \alpha \beta < 1 \)

### Step 5: Using AM-GM Inequality
Using the AM-GM inequality:
\[ \frac{\alpha + \beta + (1 - \alpha) + (1 - \beta)}{4} > \sqrt[4]{\alpha \beta (1 - \alpha)(1 - \beta)} \]
This simplifies to:
\[ \frac{2}{4} = \frac{1}{2} > \sqrt[4]{\alpha \beta (1 - \alpha)(1 - \beta)} \]

### Step 6: Finding the Minimum Value of \( b \)
From the discriminant condition:
\[ b^2 > 4ac \]
To minimize \( b \), we can set \( a = 5 \) and \( c = 1 \) (the smallest natural numbers satisfying the conditions):
\[ b^2 > 20 \implies b > \sqrt{20} \approx 4.47 \]
Thus, the smallest integer \( b \) can take is \( 5 \).

### Conclusion
The least value of \( b \) such that the quadratic equation \( ax^2 + bx + c = 0 \) has two distinct roots in the interval \( (0, 1) \) is:
\[ \boxed{5} \]