Range of `f(x)=cos^(- 1)x+2sin^(- 1)x+3tan^(- 1)x` is

Options

A. `[-(3pi)/4,(7pi)/4]`
B. `[-(5pi)/2,(7pi)/2]`
C. `[-(3pi)/2,(5pi)/2]`
D. none of these

Answer

To find the range of the function \( f(x) = \cos^{-1}(x) + 2\sin^{-1}(x) + 3\tan^{-1}(x) \), we will follow these steps:

### Step 1: Rewrite the function using known identities
We know that:
\[ \sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2} \]
Thus, we can rewrite \( f(x) \) as:
\[ f(x) = \cos^{-1}(x) + 2\sin^{-1}(x) + 3\tan^{-1}(x) = \left(\frac{\pi}{2} - \sin^{-1}(x)\right) + 2\sin^{-1}(x) + 3\tan^{-1}(x) \]
This simplifies to:
\[ f(x) = \frac{\pi}{2} + \sin^{-1}(x) + 3\tan^{-1}(x) \]

### Step 2: Determine the domain
The functions \( \sin^{-1}(x) \) and \( \tan^{-1}(x) \) have the following domains:
- \( \sin^{-1}(x) \) is defined for \( x \in [-1, 1] \)
- \( \tan^{-1}(x) \) is defined for all real numbers, but we will restrict \( x \) to the interval \( [-1, 1] \) since \( \sin^{-1}(x) \) limits the domain.

### Step 3: Analyze the behavior of the function
Both \( \sin^{-1}(x) \) and \( \tan^{-1}(x) \) are increasing functions in the interval \( [-1, 1] \). Therefore, \( f(x) \) is also an increasing function in this interval.

### Step 4: Evaluate the function at the endpoints of the interval
1. **At \( x = -1 \)**:
\[ f(-1) = \frac{\pi}{2} + \sin^{-1}(-1) + 3\tan^{-1}(-1) \]
\[ \sin^{-1}(-1) = -\frac{\pi}{2}, \quad \tan^{-1}(-1) = -\frac{\pi}{4} \]
Thus,
\[ f(-1) = \frac{\pi}{2} - \frac{\pi}{2} - \frac{3\pi}{4} = -\frac{3\pi}{4} \]

2. **At \( x = 1 \)**:
\[ f(1) = \frac{\pi}{2} + \sin^{-1}(1) + 3\tan^{-1}(1) \]
\[ \sin^{-1}(1) = \frac{\pi}{2}, \quad \tan^{-1}(1) = \frac{\pi}{4} \]
Thus,
\[ f(1) = \frac{\pi}{2} + \frac{\pi}{2} + 3 \cdot \frac{\pi}{4} = \pi + \frac{3\pi}{4} = \frac{7\pi}{4} \]

### Step 5: Determine the range
Since \( f(x) \) is increasing on the interval \( [-1, 1] \), the range of \( f(x) \) is:
\[ \text{Range} = \left[-\frac{3\pi}{4}, \frac{7\pi}{4}\right] \]

### Final Answer
The range of \( f(x) = \cos^{-1}(x) + 2\sin^{-1}(x) + 3\tan^{-1}(x) \) is:
\[ \boxed{\left[-\frac{3\pi}{4}, \frac{7\pi}{4}\right]} \]