Number of integral values of `b` for which the equation `(x^3)/3-x=b` has three distinct solutions is____

Explanation

To find the number of integral values of \( b \) for which the equation

\[ \frac{x^3}{3} - x = b \]

has three distinct solutions, we can follow these steps:

### Step 1: Define the function
Let

\[ f(x) = \frac{x^3}{3} - x - b \]

We need to find the values of \( b \) such that the equation \( f(x) = 0 \) has three distinct solutions.

### Step 2: Find the derivative
To analyze the behavior of the function, we calculate its derivative:

\[ f'(x) = x^2 - 1 \]

### Step 3: Find critical points
Setting the derivative equal to zero gives us the critical points:

\[ x^2 - 1 = 0 \implies x = -1 \text{ and } x = 1 \]

### Step 4: Analyze the sign of the derivative
We can analyze the sign of \( f'(x) \) in the intervals determined by the critical points:

- For \( x < -1 \), \( f'(x) > 0 \) (increasing)
- For \( -1 < x < 1 \), \( f'(x) < 0 \) (decreasing)
- For \( x > 1 \), \( f'(x) > 0 \) (increasing)

This indicates that \( f(x) \) has a local maximum at \( x = -1 \) and a local minimum at \( x = 1 \).

### Step 5: Evaluate \( f(x) \) at the critical points
Next, we evaluate \( f(x) \) at the critical points to find the conditions for \( b \):

1. Calculate \( f(-1) \):

\[ f(-1) = \frac{(-1)^3}{3} - (-1) - b = -\frac{1}{3} + 1 - b = \frac{2}{3} - b \]

2. Calculate \( f(1) \):

\[ f(1) = \frac{(1)^3}{3} - (1) - b = \frac{1}{3} - 1 - b = -\frac{2}{3} - b \]

### Step 6: Set up the conditions for three distinct solutions
For \( f(x) = 0 \) to have three distinct solutions, we need:

\[ f(-1) > 0 \quad \text{and} \quad f(1) < 0 \]

This leads to the inequalities:

1. \( \frac{2}{3} - b > 0 \) → \( b < \frac{2}{3} \)
2. \( -\frac{2}{3} - b < 0 \) → \( b > -\frac{2}{3} \)

### Step 7: Combine the inequalities
Combining these inequalities gives us:

\[ -\frac{2}{3} < b < \frac{2}{3} \]

### Step 8: Determine integral values of \( b \)
The integral values of \( b \) that satisfy this inequality are:

\[ b = -1, 0, 1 \]

Thus, there are **three integral values** of \( b \).

### Final Answer
The number of integral values of \( b \) for which the equation has three distinct solutions is:

\[ \boxed{3} \]