If `ax^2-bx + c=0` have two distinct roots lying in the interval `(0,1); a, b,in N`, then the least value of a , is

Options

A. 3
B. 4
C. 5
D. 6

Explanation

To find the least value of \( a \) such that the quadratic equation \( ax^2 - bx + c = 0 \) has two distinct roots lying in the interval \( (0, 1) \) where \( a, b \in \mathbb{N} \), we can follow these steps:

### Step 1: Understand the conditions for the roots
The quadratic equation \( ax^2 - bx + c = 0 \) will have two distinct roots if the discriminant \( D \) is positive. The discriminant is given by:
\[ D = b^2 - 4ac \]
For the roots to be distinct, we need:
\[ b^2 - 4ac > 0 \tag{1} \]

### Step 2: Evaluate the function at the endpoints of the interval
Let \( f(x) = ax^2 - bx + c \). Since we want the roots to lie in the interval \( (0, 1) \), we need:
\[ f(0) > 0 \quad \text{and} \quad f(1) > 0 \]
Calculating these:
\[ f(0) = c > 0 \tag{2} \]
\[ f(1) = a - b + c > 0 \tag{3} \]

### Step 3: Combine the inequalities
From inequalities (2) and (3), we have:
1. \( c > 0 \)
2. \( a - b + c > 0 \)

### Step 4: Use the AM-GM inequality
To ensure that the roots are distinct and lie in the interval \( (0, 1) \), we can apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality:
\[ \frac{\alpha + (1 - \alpha) + \beta + (1 - \beta)}{4} > \sqrt[4]{\alpha \cdot (1 - \alpha) \cdot \beta \cdot (1 - \beta)} \]
This simplifies to:
\[ \frac{1}{2} > \sqrt[4]{\alpha \beta (1 - \alpha)(1 - \beta)} \tag{4} \]

### Step 5: Establish a relationship for \( a \)
From the AM-GM inequality, we can derive:
\[ \frac{a}{16} > \alpha \beta (1 - \alpha)(1 - \beta) \]
Since \( \alpha \) and \( \beta \) are distinct roots in \( (0, 1) \), we know that:
\[ \alpha \beta (1 - \alpha)(1 - \beta) < \frac{1}{16} \]
This implies:
\[ \frac{a}{16} > \frac{1}{16} \]
Thus, we have:
\[ a > 1 \tag{5} \]

### Step 6: Find the least value of \( a \)
To satisfy all the conditions, particularly the discriminant condition \( b^2 - 4ac > 0 \), we can test integer values for \( a \):
- For \( a = 3 \): Check if there exist \( b \) and \( c \) such that all conditions hold.
- For \( a = 4 \): Similarly check.
- For \( a = 5 \): Check if conditions hold.
- For \( a = 6 \): Check.

After testing these values, we find that the least value of \( a \) that satisfies all conditions is:

\[ \boxed{5} \]