The maximum and minimum values of `f(x)=sin^(-1)x+cos^(-1)x+tan^(-1)x` respectively is

Options

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

Answer

To find the maximum and minimum values of the function \( f(x) = \sin^{-1}x + \cos^{-1}x + \tan^{-1}x \), we can follow these steps:

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

### Step 2: Determine the Domain
The domain of \( \sin^{-1}x \) and \( \cos^{-1}x \) is \( x \in [-1, 1] \). The function \( \tan^{-1}x \) is defined for all real numbers, but since \( f(x) \) is constrained by \( \sin^{-1}x \) and \( \cos^{-1}x \), we will consider \( x \in [-1, 1] \).

### Step 3: Analyze the Function
Next, we need to analyze \( f(x) \) over the interval \( [-1, 1] \). Since \( \tan^{-1}x \) is an increasing function, \( f(x) \) will also be increasing over the interval.

### Step 4: Find the Minimum Value
To find the minimum value of \( f(x) \), we evaluate it at the left endpoint of the interval:
\[ f(-1) = \frac{\pi}{2} + \tan^{-1}(-1) \]
We know that \( \tan^{-1}(-1) = -\frac{\pi}{4} \), so:
\[ f(-1) = \frac{\pi}{2} - \frac{\pi}{4} = \frac{\pi}{4} \]

### Step 5: Find the Maximum Value
To find the maximum value of \( f(x) \), we evaluate it at the right endpoint of the interval:
\[ f(1) = \frac{\pi}{2} + \tan^{-1}(1) \]
We know that \( \tan^{-1}(1) = \frac{\pi}{4} \), so:
\[ f(1) = \frac{\pi}{2} + \frac{\pi}{4} = \frac{3\pi}{4} \]

### Conclusion
Thus, the minimum value of \( f(x) \) is \( \frac{\pi}{4} \) and the maximum value is \( \frac{3\pi}{4} \).

### Final Result
- Minimum value: \( \frac{\pi}{4} \)
- Maximum value: \( \frac{3\pi}{4} \)

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