If `tan^(-)(x)/(pi)lt (pi)/(3) x in N` then the maximum vlaue of x is

Options

A. 2
B. 5
C. 7
D. None of these

Answer

To solve the problem, we need to find the maximum value of \( x \) such that:

\[ \frac{\tan^{-1}(x)}{\pi} < \frac{\pi}{3} \]

where \( x \) is a natural number.

### Step-by-Step Solution

1. **Start with the given inequality**:
\[ \frac{\tan^{-1}(x)}{\pi} < \frac{\pi}{3} \]

2. **Multiply both sides by \( \pi \)** (since \( \pi > 0 \), the direction of the inequality remains the same):
\[ \tan^{-1}(x) < \frac{\pi^2}{3} \]

3. **Take the tangent of both sides** (tangent is an increasing function):
\[ x < \tan\left(\frac{\pi^2}{3}\right) \]

4. **Calculate \( \tan\left(\frac{\pi}{3}\right) \)**:
We know that:
\[ \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \approx 1.732 \]

5. **Now, we need to find the maximum value of \( x \)**:
Since \( x \) must be less than \( \sqrt{3} \pi \), we calculate \( \sqrt{3} \pi \):
\[ \sqrt{3} \approx 1.732 \]
\[ \pi \approx 3.14 \]
\[ \sqrt{3} \pi \approx 1.732 \times 3.14 \approx 5.441 \]

6. **Since \( x \) must be a natural number**, we find the largest natural number less than \( 5.441 \):
The largest natural number less than \( 5.441 \) is \( 5 \).

### Conclusion
Thus, the maximum value of \( x \) is:
\[ \boxed{5} \]