If the range of `f(x)=tan^(1)x+2sin^(-1)x+cos^(-1)x` is `[a, b]`, then

Options

A. `a=(pi)/(4)`
B. `a=-(pi)/(2)`
C. `b=(5pi)/(4)`
D. `b=(3pi)/(2)`

Answer

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

### Step 1: Determine the Domain
The function consists of three parts: \( \tan^{-1}(x) \), \( \sin^{-1}(x) \), and \( \cos^{-1}(x) \).

- The domain of \( \tan^{-1}(x) \) is \( x \in \mathbb{R} \).
- The domain of \( \sin^{-1}(x) \) and \( \cos^{-1}(x) \) is \( x \in [-1, 1] \).

Thus, the overall domain of \( f(x) \) is \( x \in [-1, 1] \).

**Hint:** Check the individual domains of each component function to find the overall domain.

### Step 2: Rewrite the Function
We can manipulate the function to simplify it. We know that:
\[ \sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2} \]
Thus, we can rewrite \( f(x) \):
\[ f(x) = \tan^{-1}(x) + 2\sin^{-1}(x) + \cos^{-1}(x) = \tan^{-1}(x) + \sin^{-1}(x) + \frac{\pi}{2} \]

**Hint:** Use known identities to simplify the function.

### Step 3: Analyze the Behavior of the Function
Since \( f(x) \) is expressed as the sum of \( \tan^{-1}(x) \) and \( \sin^{-1}(x) \) (both of which are increasing functions), \( f(x) \) itself is an increasing function over the interval \( [-1, 1] \).

**Hint:** Understand how the monotonicity of the component functions affects the overall function.

### Step 4: Calculate the Endpoints
To find the range, we need to evaluate \( f(x) \) at the endpoints of the domain:

1. **At \( x = -1 \)**:
\[ f(-1) = \tan^{-1}(-1) + 2\sin^{-1}(-1) + \cos^{-1}(-1) \]
- \( \tan^{-1}(-1) = -\frac{\pi}{4} \)
- \( \sin^{-1}(-1) = -\frac{\pi}{2} \)
- \( \cos^{-1}(-1) = \pi \)

Thus,
\[ f(-1) = -\frac{\pi}{4} + 2(-\frac{\pi}{2}) + \pi = -\frac{\pi}{4} - \pi + \pi = -\frac{\pi}{4} \]

2. **At \( x = 1 \)**:
\[ f(1) = \tan^{-1}(1) + 2\sin^{-1}(1) + \cos^{-1}(1) \]
- \( \tan^{-1}(1) = \frac{\pi}{4} \)
- \( \sin^{-1}(1) = \frac{\pi}{2} \)
- \( \cos^{-1}(1) = 0 \)

Thus,
\[ f(1) = \frac{\pi}{4} + 2(\frac{\pi}{2}) + 0 = \frac{\pi}{4} + \pi = \frac{5\pi}{4} \]

**Hint:** Evaluate the function at the endpoints to find the minimum and maximum values.

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

Thus, we conclude:
- \( a = -\frac{\pi}{4} \)
- \( b = \frac{5\pi}{4} \)

**Final Answer:** The range of \( f(x) \) is \( \left[-\frac{\pi}{4}, \frac{5\pi}{4}\right] \).