if `-1/2 le x le 1/2 then cos^(-1)(4x^(3)-3x)` equals

Options

A. `3cos^(-1)x`
B. `2pi-3 cos^(-1)x`
C. `-2 pi + 3 cos^(-1)x`
D. `none of these

Answer

To solve the problem \( \cos^{-1}(4x^3 - 3x) \) for \( x \) in the range \( -\frac{1}{2} \leq x \leq \frac{1}{2} \), we can use the known formula relating the inverse cosine function to a polynomial expression in \( x \).

### Step-by-Step Solution:

1. **Identify the Range**: We are given that \( x \) belongs to the interval \( -\frac{1}{2} \leq x \leq \frac{1}{2} \).

2. **Recall the Formula**: The formula we will use is:
\[ \cos^{-1}(4x^3 - 3x) = \begin{cases} 3 \cos^{-1}(x) & \text{if } x \in \left[\frac{1}{2}, 1\right] \\ -2\pi + 3 \cos^{-1}(x) & \text{if } x \in \left[-\frac{1}{2}, \frac{1}{2}\right] \\ 2\pi - \cos^{-1}(4x^3 - 3x) & \text{if } x \in [-1, -\frac{1}{2}] \end{cases} \]

3. **Determine the Correct Case**: Since our \( x \) is in the range \( -\frac{1}{2} \leq x \leq \frac{1}{2} \), we will use the second case:
\[ \cos^{-1}(4x^3 - 3x) = -2\pi + 3 \cos^{-1}(x) \]

4. **Final Expression**: Therefore, we can express \( \cos^{-1}(4x^3 - 3x) \) as:
\[ \cos^{-1}(4x^3 - 3x) = -2\pi + 3 \cos^{-1}(x) \]

### Conclusion:
Thus, the final answer is:
\[ \cos^{-1}(4x^3 - 3x) = -2\pi + 3 \cos^{-1}(x) \]