If `x_(1)`, `x_(2)` and `x_(3)` are the positive roots of the equation `x^(3)-6x^(2)+3px-2p=0`, `p inR`, then the value of `sin^(-1)((1)/(x_(1))+(1)/(x_(2)))+cos^(-1)((1)/(x_(2))+(1)/(x_(3)))-tan^(-1)((1)/(x_(3))+(1)/(x_(1)))` is equal to

Options

A. `(pi)/(4)`
B. `(pi)/(2)`
C. `(3pi)/(4)`
D. `pi`

Explanation

To solve the problem, we need to analyze the given polynomial equation and the expressions involving the roots. Let's break it down step by step.

### Step 1: Understand the Given Equation
The equation is given as:
\[ x^3 - 6x^2 + 3px - 2p = 0 \]
where \(x_1\), \(x_2\), and \(x_3\) are the positive roots.

### Step 2: Apply Vieta's Formulas
From Vieta's formulas, we know:
- \(x_1 + x_2 + x_3 = 6\)
- \(x_1 x_2 + x_2 x_3 + x_3 x_1 = 3p\)
- \(x_1 x_2 x_3 = 2p\)

### Step 3: Express \(p\) in Terms of Roots
From the equations above, we can express \(p\) in terms of the roots:
1. From \(x_1 + x_2 + x_3 = 6\), we have:
\[ x_1 + x_2 + x_3 = 6 \quad \text{(1)} \]
2. From \(x_1 x_2 x_3 = 2p\), we can express \(p\):
\[ p = \frac{x_1 x_2 x_3}{2} \quad \text{(2)} \]
3. From \(x_1 x_2 + x_2 x_3 + x_3 x_1 = 3p\), we can express \(p\) again:
\[ p = \frac{x_1 x_2 + x_2 x_3 + x_3 x_1}{3} \quad \text{(3)} \]

### Step 4: Analyze the Roots
Since \(x_1\), \(x_2\), and \(x_3\) are positive, we can assume \(x_1 = x_2 = x_3 = 2\) (as a potential solution that satisfies \(x_1 + x_2 + x_3 = 6\)).

### Step 5: Calculate \(p\)
Substituting \(x_1 = x_2 = x_3 = 2\) into equation (2):
\[ p = \frac{2 \cdot 2 \cdot 2}{2} = 4 \]

### Step 6: Substitute Values into the Expression
Now we need to evaluate:
\[ \sin^{-1}\left(\frac{1}{x_1} + \frac{1}{x_2}\right) + \cos^{-1}\left(\frac{1}{x_2} + \frac{1}{x_3}\right) - \tan^{-1}\left(\frac{1}{x_3} + \frac{1}{x_1}\right) \]
Substituting \(x_1 = x_2 = x_3 = 2\):
1. \(\frac{1}{x_1} + \frac{1}{x_2} = \frac{1}{2} + \frac{1}{2} = 1\)
2. \(\frac{1}{x_2} + \frac{1}{x_3} = \frac{1}{2} + \frac{1}{2} = 1\)
3. \(\frac{1}{x_3} + \frac{1}{x_1} = \frac{1}{2} + \frac{1}{2} = 1\)

### Step 7: Evaluate the Inverse Functions
Now substituting these values:
1. \(\sin^{-1}(1) = \frac{\pi}{2}\)
2. \(\cos^{-1}(1) = 0\)
3. \(\tan^{-1}(1) = \frac{\pi}{4}\)

### Step 8: Combine the Results
Now we combine the results:
\[ \frac{\pi}{2} + 0 - \frac{\pi}{4} = \frac{\pi}{2} - \frac{\pi}{4} = \frac{\pi}{4} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{\pi}{4} \]