Find the range of `f(x) = (sin^(-1) x)^(2) + 2pi cos^(-1) x + pi^(2)`

Options

A. `[(pi^2)/4, (9pi^2)/4 ]`
B. `[(3pi^2)/4, (11pi^2)/4 ]`
C. `[(5pi^2)/4, (13pi^2)/4 ]`
D. `[(7pi^2)/4, (15pi^2)/4 ]`

Explanation

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

### Step 1: Rewrite the Function
We can rewrite \( \cos^{-1} x \) in terms of \( \sin^{-1} x \):
\[ \cos^{-1} x = \frac{\pi}{2} - \sin^{-1} x \]
Thus, we can express \( f(x) \) as:
\[ f(x) = (\sin^{-1} x)^2 + 2\pi \left(\frac{\pi}{2} - \sin^{-1} x\right) + \pi^2 \]

### Step 2: Simplify the Function
Substituting the expression for \( \cos^{-1} x \) into \( f(x) \):
\[ f(x) = (\sin^{-1} x)^2 + \pi^2 - 2\pi \sin^{-1} x + \pi^2 \]
This simplifies to:
\[ f(x) = (\sin^{-1} x)^2 - 2\pi \sin^{-1} x + 2\pi^2 \]

### Step 3: Identify the Quadratic Form
Notice that the expression can be rewritten in a quadratic form:
\[ f(x) = (\sin^{-1} x - \pi)^2 + \pi^2 \]
This indicates that \( f(x) \) is a quadratic function in terms of \( \sin^{-1} x \).

### Step 4: Determine the Range of \( \sin^{-1} x \)
The function \( \sin^{-1} x \) is defined for \( x \in [-1, 1] \). Therefore, the range of \( \sin^{-1} x \) is:
\[ \sin^{-1} x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \]

### Step 5: Find the Minimum and Maximum Values of \( f(x) \)
To find the minimum and maximum values of \( f(x) \), we evaluate \( f(x) \) at the endpoints of the interval for \( \sin^{-1} x \).

1. **Minimum at \( \sin^{-1} x = -\frac{\pi}{2} \)**:
\[ f\left(-\frac{\pi}{2}\right) = \left(-\frac{\pi}{2} - \pi\right)^2 + \pi^2 = \left(-\frac{3\pi}{2}\right)^2 + \pi^2 = \frac{9\pi^2}{4} + \pi^2 = \frac{13\pi^2}{4} \]

2. **Maximum at \( \sin^{-1} x = \frac{\pi}{2} \)**:
\[ f\left(\frac{\pi}{2}\right) = \left(\frac{\pi}{2} - \pi\right)^2 + \pi^2 = \left(-\frac{\pi}{2}\right)^2 + \pi^2 = \frac{\pi^2}{4} + \pi^2 = \frac{5\pi^2}{4} \]

### Step 6: Conclusion on the Range
Thus, the range of \( f(x) \) is:
\[ \left[\frac{5\pi^2}{4}, \frac{13\pi^2}{4}\right] \]