If the equation `x^(3 )-3x+a=0` has distinct roots between 0 and 1, then the value of a is

Options

A. `+ ive`
B. `-ive`
C. 2
D. Does not exist

Answer

To solve the equation \( x^3 - 3x + a = 0 \) for the value of \( a \) such that the equation has distinct roots between 0 and 1, we can follow these steps:

### Step 1: Identify the function
We start by defining the function based on the given equation:
\[ f(x) = x^3 - 3x + a \]

### Step 2: Find the derivative
To find the critical points where the function changes direction, we need to calculate the derivative:
\[ f'(x) = 3x^2 - 3 \]

### Step 3: Set the derivative to zero
To find the critical points, we set the derivative equal to zero:
\[ 3x^2 - 3 = 0 \]
\[ x^2 = 1 \implies x = 1 \text{ or } x = -1 \]

### Step 4: Analyze the critical points
Since we are interested in the interval \( (0, 1) \), we only consider \( x = 1 \) as a critical point. We need to check the behavior of \( f(x) \) around this point.

### Step 5: Evaluate the function at the endpoints
Next, we evaluate the function at the endpoints of the interval:
- At \( x = 0 \):
\[ f(0) = 0^3 - 3(0) + a = a \]
- At \( x = 1 \):
\[ f(1) = 1^3 - 3(1) + a = 1 - 3 + a = a - 2 \]

### Step 6: Ensure distinct roots
For the cubic equation to have distinct roots between 0 and 1, \( f(0) \) and \( f(1) \) must have opposite signs:
1. \( f(0) = a \)
2. \( f(1) = a - 2 \)

Thus, we need:
\[ a > 0 \quad \text{(for } f(0) > 0\text{)} \]
\[ a - 2 < 0 \quad \text{(for } f(1) < 0\text{)} \implies a < 2 \]

### Step 7: Combine the inequalities
From the above inequalities, we can combine them:
\[ 0 < a < 2 \]

### Step 8: Conclusion
The value of \( a \) must be in the interval \( (0, 2) \) for the equation \( x^3 - 3x + a = 0 \) to have distinct roots between 0 and 1.