For `x, y, z, t in R, sin^(-1) x + cos^(-1) y + sec^(-1) z ge t^(2) - sqrt(2pi t) + 3pi`
The value of `cos^(-1) ("min" {x, y, z})` is

Options

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

Explanation

To solve the problem, we need to analyze the given inequality and find the value of \( \cos^{-1}(\min\{x, y, z\}) \).

### Step-by-Step Solution:

1. **Understanding the Given Inequality**:
We start with the inequality:
\[ \sin^{-1}(x) + \cos^{-1}(y) + \sec^{-1}(z) \geq t^2 - \sqrt{2\pi t} + 3\pi \]

2. **Identifying the Ranges**:
- The range of \( \sin^{-1}(x) \) is \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \).
- The range of \( \cos^{-1}(y) \) is \( [0, \pi] \).
- The range of \( \sec^{-1}(z) \) is \( [0, \frac{\pi}{2}) \cup (\frac{\pi}{2}, \pi] \).

3. **Finding Maximum Values**:
The maximum values of each function are:
- \( \sin^{-1}(x) \) can be at most \( \frac{\pi}{2} \).
- \( \cos^{-1}(y) \) can be at most \( \pi \).
- \( \sec^{-1}(z) \) can be at most \( \pi \).

Therefore, the maximum possible sum is:
\[ \sin^{-1}(x) + \cos^{-1}(y) + \sec^{-1}(z) \leq \frac{\pi}{2} + \pi + \pi = \frac{5\pi}{2} \]

4. **Setting Up the Inequality**:
From the inequality, we can write:
\[ t^2 - \sqrt{2\pi t} + 3\pi \leq \frac{5\pi}{2} \]

5. **Rearranging the Inequality**:
Rearranging gives us:
\[ t^2 - \sqrt{2\pi t} + 3\pi - \frac{5\pi}{2} \leq 0 \]
Simplifying this:
\[ t^2 - \sqrt{2\pi t} + \frac{1\pi}{2} \leq 0 \]

6. **Completing the Square**:
We can complete the square for the quadratic in \( t \):
\[ \left(t - \frac{\sqrt{2\pi}}{2}\right)^2 - \frac{2\pi}{4} + \frac{\pi}{2} \leq 0 \]
This simplifies to:
\[ \left(t - \frac{\sqrt{2\pi}}{2}\right)^2 \leq 0 \]
This implies:
\[ t - \frac{\sqrt{2\pi}}{2} = 0 \implies t = \frac{\sqrt{2\pi}}{2} \]

7. **Finding Values of \( x, y, z \)**:
To satisfy the original inequality, we can set:
- \( \sin^{-1}(x) = \frac{\pi}{2} \) which gives \( x = 1 \).
- \( \cos^{-1}(y) = \pi \) which gives \( y = -1 \).
- \( \sec^{-1}(z) = \pi \) which gives \( z = -1 \).

8. **Finding the Minimum**:
Now we find \( \min\{x, y, z\} = \min\{1, -1, -1\} = -1 \).

9. **Calculating \( \cos^{-1}(\min\{x, y, z\}) \)**:
Finally, we compute:
\[ \cos^{-1}(-1) = \pi \]

### Conclusion:
The value of \( \cos^{-1}(\min\{x, y, z\}) \) is \( \pi \).