If `f(x) = sin^(-1) x. cos^(-1) x. tan^(-1) x . cot^(-1) x. sec^(-1) x. cosec^(-1) x`, then which of the following statement (s) hold(s) good?

Options

A. (a) The graph of `y = f(x)` does not lie above `x`-axis
B. (b) The non-negative difference between the maximum and the minimum value of the function ` y = f (x) " is " (3 pi^(6))/64`
C. (c) The function `y = f(x)` is not injective.
D. (d) Number of non-negative integers in the domain of `f(x)` is 2.

Answer

To solve the problem, we need to analyze the function \( f(x) = \sin^{-1} x \cdot \cos^{-1} x \cdot \tan^{-1} x \cdot \cot^{-1} x \cdot \sec^{-1} x \cdot \csc^{-1} x \) and determine which statements about it are true.

### Step 1: Determine the Domain of \( f(x) \)

1. **Identify the domains of each inverse trigonometric function:**
- \( \sin^{-1} x \) and \( \cos^{-1} x \) have a domain of \( [-1, 1] \).
- \( \tan^{-1} x \) and \( \cot^{-1} x \) have a domain of \( (-\infty, \infty) \).
- \( \sec^{-1} x \) and \( \csc^{-1} x \) have a domain of \( (-\infty, -1] \cup [1, \infty) \).

2. **Find the intersection of these domains:**
- The intersection of \( [-1, 1] \) (from \( \sin^{-1} x \) and \( \cos^{-1} x \)) and \( (-\infty, -1] \cup [1, \infty) \) (from \( \sec^{-1} x \) and \( \csc^{-1} x \)) is only the points \( -1 \) and \( 1 \).

**Conclusion:** The domain of \( f(x) \) is \( \{-1, 1\} \).

### Step 2: Evaluate \( f(x) \) at the Domain Points

1. **Calculate \( f(1) \):**
- \( \sin^{-1}(1) = \frac{\pi}{2} \)
- \( \cos^{-1}(1) = 0 \)
- \( \tan^{-1}(1) = \frac{\pi}{4} \)
- \( \cot^{-1}(1) = \frac{\pi}{4} \)
- \( \sec^{-1}(1) = 0 \)
- \( \csc^{-1}(1) = \frac{\pi}{2} \)

Therefore,
\[ f(1) = \sin^{-1}(1) \cdot \cos^{-1}(1) \cdot \tan^{-1}(1) \cdot \cot^{-1}(1) \cdot \sec^{-1}(1) \cdot \csc^{-1}(1) = \frac{\pi}{2} \cdot 0 \cdot \frac{\pi}{4} \cdot \frac{\pi}{4} \cdot 0 \cdot \frac{\pi}{2} = 0 \]

2. **Calculate \( f(-1) \):**
- \( \sin^{-1}(-1) = -\frac{\pi}{2} \)
- \( \cos^{-1}(-1) = \pi \)
- \( \tan^{-1}(-1) = -\frac{\pi}{4} \)
- \( \cot^{-1}(-1) = \frac{3\pi}{4} \)
- \( \sec^{-1}(-1) = \pi \)
- \( \csc^{-1}(-1) = -\frac{\pi}{2} \)

Therefore,
\[ f(-1) = -\frac{\pi}{2} \cdot \pi \cdot -\frac{\pi}{4} \cdot \frac{3\pi}{4} \cdot \pi \cdot -\frac{\pi}{2} \]
Simplifying this gives:
\[ f(-1) = \left(-\frac{\pi}{2} \cdot -\frac{\pi}{2}\right) \cdot \left(\pi \cdot \pi\right) \cdot \left(-\frac{\pi}{4} \cdot \frac{3\pi}{4}\right) = \frac{3\pi^6}{64} \]

### Step 3: Analyze the Results

1. **Maximum and Minimum Values:**
- The maximum value of \( f(x) \) is \( f(1) = 0 \).
- The minimum value of \( f(x) \) is \( f(-1) = -\frac{3\pi^6}{64} \).

2. **Non-negative difference:**
\[ \text{Non-negative difference} = 0 - \left(-\frac{3\pi^6}{64}\right) = \frac{3\pi^6}{64} \]

### Step 4: Check the Statements

- **Statement A:** The graph of \( f(x) \) does not lie above the x-axis. **True** (since maximum is 0).
- **Statement B:** The non-negative difference between the maximum and minimum value of the function \( y \) is equal to \( \frac{3\pi^6}{64} \). **True**.
- **Statement C:** The function \( f(x) \) is not injective. **False** (since it only takes two values).
- **Statement D:** The number of non-negative integers in the domain of \( f(x) \) is 2. **False** (only 1 non-negative integer, which is 1).

### Final Conclusion

The correct statements are A and B.

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