For `f:N rarr R f(x)=(x^(3)-11^(3))^(1//3)`, the number of positive integral values f(x) can take is

Options

A. 1
B. 2
C. 3
D. 0

Answer

To solve the problem, we need to analyze the function \( f(x) = (x^3 - 11^3)^{1/3} \) and determine the number of positive integral values it can take.

### Step-by-step Solution:

1. **Rewrite the Function**:
The function can be rewritten using the difference of cubes formula:
\[ f(x) = (x^3 - 11^3)^{1/3} = (x - 11)(x^2 + 11x + 121)^{1/3} \]

2. **Identify Conditions for Positive Integral Values**:
For \( f(x) \) to be a positive integer, \( (x^3 - 11^3) \) must be a positive perfect cube. This implies:
\[ x^3 - 11^3 > 0 \implies x > 11 \]

3. **Set Up the Perfect Cube Condition**:
Let \( k \) be a positive integer such that:
\[ x^3 - 11^3 = k^3 \]
Rearranging gives:
\[ x^3 = k^3 + 11^3 \]

4. **Express \( x \)**:
Taking the cube root of both sides:
\[ x = (k^3 + 11^3)^{1/3} \]

5. **Determine the Range of \( k \)**:
Since \( x \) must be greater than 11, we can set up the inequality:
\[ (k^3 + 11^3)^{1/3} > 11 \]
Cubing both sides results in:
\[ k^3 + 11^3 > 11^3 \implies k^3 > 0 \implies k > 0 \]

6. **Finding Positive Integral Values**:
The values of \( f(x) \) can be expressed as:
\[ f(x) = k \]
where \( k \) is a positive integer. The function \( f(x) \) can take any positive integer value as long as \( k \) is a positive integer.

7. **Conclusion**:
Since \( k \) can take any positive integer value, the number of positive integral values \( f(x) \) can take is infinite.

### Final Answer:
The number of positive integral values \( f(x) \) can take is **infinite**.