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

Options

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

Answer

To solve the problem, we need to evaluate \( \cos^{-1}(4x^3 - 3x) \) given that \( \frac{1}{2} \leq x \leq 1 \).

### Step-by-step Solution:

1. **Substitution**: Let \( x = \cos \theta \).
- Since \( \frac{1}{2} \leq x \leq 1 \), it follows that \( \theta \) will be in the range \( \frac{\pi}{3} \leq \theta \leq 0 \).

2. **Rewrite the expression**: Substitute \( x \) in the expression:
\[ \cos^{-1}(4x^3 - 3x) = \cos^{-1}(4(\cos \theta)^3 - 3(\cos \theta)) \]

3. **Use the cosine triple angle formula**: We know that:
\[ \cos 3\theta = 4 \cos^3 \theta - 3 \cos \theta \]
Therefore, we can rewrite the expression as:
\[ \cos^{-1}(4 \cos^3 \theta - 3 \cos \theta) = \cos^{-1}(\cos 3\theta) \]

4. **Apply the inverse cosine property**: The property \( \cos^{-1}(\cos x) = x \) holds for \( x \) in the range of \( [0, \pi] \). Since \( 3\theta \) will also lie in this range:
\[ \cos^{-1}(\cos 3\theta) = 3\theta \]

5. **Substitute back for \( \theta \)**: Recall that \( \theta = \cos^{-1}(x) \):
\[ 3\theta = 3 \cos^{-1}(x) \]

### Final Answer:
Thus, we conclude that:
\[ \cos^{-1}(4x^3 - 3x) = 3 \cos^{-1}(x) \]