The number of non-zero integral solution of K for which the equation `(x^(3))/(3) - 4x= K` has three distinct solution is

Answer

To solve the problem, we need to find the number of non-zero integral solutions for \( K \) such that the equation

\[ \frac{x^3}{3} - 4x = K \]

has three distinct solutions.

### Step 1: Define the function
Let

\[ f(x) = \frac{x^3}{3} - 4x \]

### Step 2: Differentiate the function
We differentiate \( f(x) \) to find its critical points:

\[ f'(x) = \frac{d}{dx}\left(\frac{x^3}{3}\right) - \frac{d}{dx}(4x) = x^2 - 4 \]

### Step 3: Find critical points
Set the derivative equal to zero to find the critical points:

\[ x^2 - 4 = 0 \]

This gives us:

\[ x^2 = 4 \implies x = -2 \text{ and } x = 2 \]

### Step 4: Analyze the sign of the derivative
To determine the nature of these critical points, we analyze the sign of \( f'(x) \):

- For \( x < -2 \): \( f'(x) > 0 \) (increasing)
- For \( -2 < x < 2 \): \( f'(x) < 0 \) (decreasing)
- For \( x > 2 \): \( f'(x) > 0 \) (increasing)

This indicates that \( x = -2 \) is a local maximum and \( x = 2 \) is a local minimum.

### Step 5: Evaluate the function at critical points
Next, we evaluate \( f(x) \) at the critical points:

\[ f(-2) = \frac{(-2)^3}{3} - 4(-2) = \frac{-8}{3} + 8 = \frac{-8 + 24}{3} = \frac{16}{3} \]

\[ f(2) = \frac{(2)^3}{3} - 4(2) = \frac{8}{3} - 8 = \frac{8 - 24}{3} = \frac{-16}{3} \]

### Step 6: Determine the range for \( K \)
For the function \( f(x) \) to have three distinct solutions, \( K \) must lie between the maximum and minimum values:

\[ f(2) < K < f(-2) \implies \frac{-16}{3} < K < \frac{16}{3} \]

### Step 7: Find the integer solutions
Now we need to find the integer values of \( K \) that satisfy:

\[ -5.33 < K < 5.33 \]

The integers in this range are:

\[ -5, -4, -3, -2, -1, 1, 2, 3, 4, 5 \]

### Step 8: Count the non-zero integral solutions
The non-zero integers are:

\[ -5, -4, -3, -2, -1, 1, 2, 3, 4, 5 \]

This gives us a total of 10 non-zero integral solutions for \( K \).

### Final Answer
Thus, the number of non-zero integral solutions for \( K \) is

\[ \boxed{10} \]