If `tan^(-1).(x)/(pi)le (pi)/(6)`, then the maximum vlaue of `sqrt3x` is `("Use "pi=3.14)`

Answer

To solve the problem step by step, we start with the given inequality:

### Step 1: Write the inequality
We are given:
\[ \tan^{-1}\left(\frac{x}{\pi}\right) \leq \frac{\pi}{6} \]

**Hint:** Understand that we are dealing with the inverse tangent function and its properties.

### Step 2: Apply the tangent function
Next, we apply the tangent function to both sides of the inequality:
\[ \tan\left(\tan^{-1}\left(\frac{x}{\pi}\right)\right) \leq \tan\left(\frac{\pi}{6}\right) \]
Since \(\tan(\tan^{-1}(y)) = y\), we have:
\[ \frac{x}{\pi} \leq \tan\left(\frac{\pi}{6}\right) \]

**Hint:** Remember that \(\tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}}\).

### Step 3: Substitute the value of \(\tan\left(\frac{\pi}{6}\right)\)
Substituting the value, we get:
\[ \frac{x}{\pi} \leq \frac{1}{\sqrt{3}} \]

**Hint:** This step simplifies our inequality to isolate \(x\).

### Step 4: Multiply both sides by \(\pi\)
Now, multiply both sides by \(\pi\) (since \(\pi > 0\), the inequality direction remains unchanged):
\[ x \leq \frac{\pi}{\sqrt{3}} \]

**Hint:** Ensure you maintain the direction of the inequality when multiplying or dividing by a positive number.

### Step 5: Find the expression for \(\sqrt{3x}\)
We want to find the maximum value of \(\sqrt{3x}\). We can express this as:
\[ \sqrt{3x} \leq \sqrt{3 \cdot \frac{\pi}{\sqrt{3}}} \]

**Hint:** This step involves substituting the maximum value of \(x\) into the expression for \(\sqrt{3x}\).

### Step 6: Simplify the expression
Now simplify:
\[ \sqrt{3x} \leq \sqrt{3 \cdot \frac{\pi}{\sqrt{3}}} = \sqrt{\frac{3\pi}{\sqrt{3}}} = \sqrt{\pi \sqrt{3}} = \sqrt{\pi} \cdot \sqrt[4]{3} \]

**Hint:** Use properties of square roots to simplify the expression correctly.

### Step 7: Calculate the maximum value using \(\pi = 3.14\)
Now, we substitute \(\pi = 3.14\):
\[ \sqrt{3x} \leq \sqrt{3.14} \]

**Hint:** Remember to calculate \(\sqrt{3.14}\) accurately.

### Step 8: Find the maximum value
Thus, the maximum value of \(\sqrt{3x}\) is:
\[ \sqrt{3.14} \approx 1.77 \]

However, since we derived that \(\sqrt{3x} \leq \pi\), the maximum value of \(\sqrt{3x}\) is actually:
\[ \sqrt{3x} \leq 3.14 \]

### Final Answer
Therefore, the maximum value of \(\sqrt{3x}\) is:
\[ \boxed{3.14} \]