The set of values of 'a' for which one negative and two positive roots of the equation `x^(3) - 3x + a = 0` are possible, is

Options

A. (0, 2)
B. (0, 4)
C. (2, 4)
D. (0, 10)

Explanation

To find the set of values of 'a' for which the equation \( x^3 - 3x + a = 0 \) has one negative and two positive roots, we can follow these steps:

### Step 1: Define the functions
Let \( f(x) = x^3 - 3x \) and \( g(x) = -a \). The roots of the equation \( x^3 - 3x + a = 0 \) occur where the graphs of \( f(x) \) and \( g(x) \) intersect.

### Step 2: Analyze the function \( f(x) \)
To analyze the function \( f(x) \), we first find its critical points by taking the derivative:
\[ f'(x) = 3x^2 - 3 \]
Setting the derivative equal to zero to find critical points:
\[ 3x^2 - 3 = 0 \implies x^2 = 1 \implies x = \pm 1 \]

### Step 3: Find the values of \( f(x) \) at critical points
Now, we evaluate \( f(x) \) at these critical points:
- For \( x = 1 \):
\[ f(1) = 1^3 - 3(1) = 1 - 3 = -2 \]
- For \( x = -1 \):
\[ f(-1) = (-1)^3 - 3(-1) = -1 + 3 = 2 \]

### Step 4: Determine the behavior of \( f(x) \)
Next, we find the limits of \( f(x) \) as \( x \) approaches infinity and negative infinity:
- As \( x \to -\infty \), \( f(x) \to -\infty \)
- As \( x \to \infty \), \( f(x) \to \infty \)

### Step 5: Identify the range of \( f(x) \)
From the critical points and their corresponding values, we see that:
- The maximum value of \( f(x) \) is 2 (at \( x = -1 \))
- The minimum value of \( f(x) \) is -2 (at \( x = 1 \))

### Step 6: Analyze the intersection with \( g(x) = -a \)
For the equation to have one negative and two positive roots, the horizontal line \( g(x) = -a \) must intersect the graph of \( f(x) \) in such a way that:
- It must be below the maximum point (i.e., \( -a < 2 \)) and above the minimum point (i.e., \( -a > -2 \)).

This gives us the inequalities:
\[ -2 < -a < 2 \]

### Step 7: Solve the inequalities
Multiplying through by -1 (and reversing the inequalities):
\[ 2 > a > 0 \]
Thus, we can conclude that:
\[ 0 < a < 2 \]

### Final Answer
The set of values of \( a \) for which the equation has one negative and two positive roots is:
\[ (0, 2) \]