Range of `f(x)=sin^(-1)[x-1]+2cos^(-1)[x-2]` ([.] denotes greatest integer function)

Options

A. `{-(pi)/(2), 0}`
B. `{(pi)/(2), 2pi}`
C. `{(pi)/(4), (pi)/(2)}`
D. `{(3pi)/(2), 2pi}`

Answer

To find the range of the function \( f(x) = \sin^{-1}(x-1) + 2\cos^{-1}(x-2) \), we will analyze the components of the function step by step.

### Step 1: Determine the domain of the function
The function involves the inverse sine and inverse cosine functions.

- The domain of \( \sin^{-1}(x-1) \) is \( -1 \leq x - 1 \leq 1 \), which simplifies to \( 0 \leq x \leq 2 \).
- The domain of \( \cos^{-1}(x-2) \) is \( -1 \leq x - 2 \leq 1 \), which simplifies to \( 1 \leq x \leq 3 \).

The overall domain of \( f(x) \) is the intersection of these two intervals:
\[ [1, 2] \]

### Step 2: Analyze \( f(x) \) within the domain
Now, we will evaluate \( f(x) \) at the endpoints of the domain and check for any critical points within the interval.

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

2. **At \( x = 2 \)**:
\[ f(2) = \sin^{-1}(2-1) + 2\cos^{-1}(2-2) = \sin^{-1}(1) + 2\cos^{-1}(0) = \frac{\pi}{2} + 2\left(\frac{\pi}{2}\right) = \frac{\pi}{2} + \pi = \frac{3\pi}{2} \]

### Step 3: Determine the behavior of \( f(x) \) in the interval
To find if there are any critical points, we can differentiate \( f(x) \):
\[ f'(x) = \frac{d}{dx}[\sin^{-1}(x-1)] + 2\frac{d}{dx}[\cos^{-1}(x-2)] \]
Using the derivatives:
\[ \frac{d}{dx}[\sin^{-1}(u)] = \frac{1}{\sqrt{1-u^2}} \cdot \frac{du}{dx}, \quad \text{and} \quad \frac{d}{dx}[\cos^{-1}(u)] = -\frac{1}{\sqrt{1-u^2}} \cdot \frac{du}{dx} \]
where \( u = x-1 \) for sine and \( u = x-2 \) for cosine.

Calculating these derivatives:
\[ f'(x) = \frac{1}{\sqrt{1-(x-1)^2}} - \frac{2}{\sqrt{1-(x-2)^2}} \]

### Step 4: Evaluate the critical points
We check \( f'(x) \) at \( x = 1 \) and \( x = 2 \) to see if it changes sign:
- At \( x = 1 \), \( f'(1) \) is positive.
- At \( x = 2 \), \( f'(2) \) is negative.

This indicates that \( f(x) \) is decreasing in the interval \( [1, 2] \).

### Step 5: Conclusion
The maximum value of \( f(x) \) occurs at \( x = 1 \) and is \( 2\pi \), while the minimum value occurs at \( x = 2 \) and is \( \frac{3\pi}{2} \).

Thus, the range of \( f(x) \) is:
\[ \left[ \frac{3\pi}{2}, 2\pi \right] \]