If `x_1,x_2,x_3,x_4,x_5,x_6` all are independent then the maximum and minimum values of `[sin^(-1)x_1]+[cos^(-1)x_2]+[tan^(-1)x_3]+[cot^(-1)x_4]+[sec^(-1)x_5]+[cosec^(-1)x_6]`, where [] represents greatest integer function, respectively are

Options

A. 9,3
B. 11,5
C. 12,-6
D. 12,-3

Answer

To find the maximum and minimum values of the expression

\[ [\sin^{-1} x_1] + [\cos^{-1} x_2] + [\tan^{-1} x_3] + [\cot^{-1} x_4] + [\sec^{-1} x_5] + [\csc^{-1} x_6] \]

where \([]\) denotes the greatest integer function, we will analyze the ranges of each of the inverse trigonometric functions involved.

### Step 1: Determine the ranges of each function

1. **Range of \(\sin^{-1} x\)**: The range is \([- \frac{\pi}{2}, \frac{\pi}{2}]\).
2. **Range of \(\cos^{-1} x\)**: The range is \([0, \pi]\).
3. **Range of \(\tan^{-1} x\)**: The range is \([- \frac{\pi}{2}, \frac{\pi}{2}]\).
4. **Range of \(\cot^{-1} x\)**: The range is \((0, \pi)\).
5. **Range of \(\sec^{-1} x\)**: The range is \([0, \frac{\pi}{2}) \cup (\frac{\pi}{2}, \pi]\).
6. **Range of \(\csc^{-1} x\)**: The range is \([- \frac{\pi}{2}, 0) \cup (0, \frac{\pi}{2}]\).

### Step 2: Calculate the maximum values

To find the maximum value of the entire expression, we take the maximum of each individual function:

- Maximum of \([\sin^{-1} x_1]\) is \([\frac{\pi}{2}] = 1\).
- Maximum of \([\cos^{-1} x_2]\) is \([\pi] = 3\).
- Maximum of \([\tan^{-1} x_3]\) is \([\frac{\pi}{2}] = 1\).
- Maximum of \([\cot^{-1} x_4]\) is \([\pi] = 3\).
- Maximum of \([\sec^{-1} x_5]\) is \([\pi] = 3\).
- Maximum of \([\csc^{-1} x_6]\) is \([\frac{\pi}{2}] = 1\).

Adding these maximum values together gives:

\[ 1 + 3 + 1 + 3 + 3 + 1 = 12 \]

### Step 3: Calculate the minimum values

To find the minimum value of the entire expression, we take the minimum of each individual function:

- Minimum of \([\sin^{-1} x_1]\) is \([- \frac{\pi}{2}] = -2\).
- Minimum of \([\cos^{-1} x_2]\) is \([0] = 0\).
- Minimum of \([\tan^{-1} x_3]\) is \([- \frac{\pi}{2}] = -2\).
- Minimum of \([\cot^{-1} x_4]\) is \([0] = 0\).
- Minimum of \([\sec^{-1} x_5]\) is \([0] = 0\).
- Minimum of \([\csc^{-1} x_6]\) is \([- \frac{\pi}{2}] = -2\).

Adding these minimum values together gives:

\[ -2 + 0 - 2 + 0 + 0 - 2 = -6 \]

### Final Results

Thus, the maximum value of the expression is **12** and the minimum value is **-6**.