If `f(x)=tan^(-1)((sqrt(3)x-3x)/(3sqrt(3)+x^(2)))+tan^(-1)(x/(sqrt(3))),0lexle3,` then range of `f(x)` is

Options

A. `[0,(pi)/2)`
B. `[0,(pi)/4]`
C. `[(pi)/6,(pi)/3]`
D. `[0,(pi)/3]`

Explanation

To find the range of the function \( f(x) = \tan^{-1}\left(\frac{\sqrt{3}x - 3x}{3\sqrt{3} + x^2}\right) + \tan^{-1}\left(\frac{x}{\sqrt{3}}\right) \) for \( 0 \leq x \leq 3 \), we can follow these steps:

### Step 1: Simplify the function
First, we can rewrite the first term of \( f(x) \):
\[ f(x) = \tan^{-1}\left(\frac{\sqrt{3}x - 3x}{3\sqrt{3} + x^2}\right) + \tan^{-1}\left(\frac{x}{\sqrt{3}}\right) \]

### Step 2: Use the formula for the sum of arctangents
We can apply the formula for the sum of arctangents:
\[ \tan^{-1}(A) + \tan^{-1}(B) = \tan^{-1}\left(\frac{A + B}{1 - AB}\right) \]
where \( A = \frac{\sqrt{3}x - 3x}{3\sqrt{3} + x^2} \) and \( B = \frac{x}{\sqrt{3}} \).

### Step 3: Find \( A + B \) and \( 1 - AB \)
Calculating \( A + B \):
\[ A + B = \frac{\sqrt{3}x - 3x}{3\sqrt{3} + x^2} + \frac{x}{\sqrt{3}} \]

To combine these fractions, we need a common denominator:
\[ A + B = \frac{(\sqrt{3}x - 3x)\sqrt{3} + x(3\sqrt{3} + x^2)}{(3\sqrt{3} + x^2)\sqrt{3}} \]

Calculating \( 1 - AB \):
\[ AB = \left(\frac{\sqrt{3}x - 3x}{3\sqrt{3} + x^2}\right) \cdot \left(\frac{x}{\sqrt{3}}\right) \]

### Step 4: Substitute back into the arctangent formula
Now we can substitute \( A + B \) and \( 1 - AB \) back into the arctangent formula.

### Step 5: Analyze the function
Since \( \tan^{-1}(x) \) is an increasing function, we can find the minimum and maximum values of \( f(x) \) by evaluating it at the endpoints of the interval \( [0, 3] \).

### Step 6: Calculate \( f(0) \) and \( f(3) \)
1. **At \( x = 0 \)**:
\[ f(0) = \tan^{-1}(0) + \tan^{-1}(0) = 0 \]

2. **At \( x = 3 \)**:
\[ f(3) = \tan^{-1}\left(\frac{\sqrt{3}(3) - 3(3)}{3\sqrt{3} + 3^2}\right) + \tan^{-1}\left(\frac{3}{\sqrt{3}}\right) \]
Simplifying the first term:
\[ f(3) = \tan^{-1}\left(\frac{3\sqrt{3} - 9}{3\sqrt{3} + 9}\right) + \tan^{-1}(\sqrt{3}) \]
The first term simplifies to \( \tan^{-1}(0) = 0 \) and the second term is \( \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \).

### Step 7: Determine the range
Thus, the function \( f(x) \) ranges from \( 0 \) to \( \frac{\pi}{4} \).

### Final Answer
The range of \( f(x) \) is:
\[ \text{Range of } f(x) = [0, \frac{\pi}{4}] \]