Let `cos^(-1) (4x^(3) -3x) = a + b cos^(-1) x`
If `x in ((1)/(2), 1]`, then the value of `lim_( y to a)b cos (y)` is

Options

A. `-1//3`
B. `-3`
C. `(1)/(3)`
D. 3

Explanation

To solve the problem, we need to find the limit \( \lim_{y \to a} b \cos(y) \) given that \( \cos^{-1}(4x^3 - 3x) = a + b \cos^{-1}(x) \) for \( x \in \left( \frac{1}{2}, 1 \right] \).

### Step-by-Step Solution:

1. **Substitute \( x \) with \( \cos(\theta) \)**:
Let \( x = \cos(\theta) \). Then, we have:
\[ \cos^{-1}(4x^3 - 3x) = \cos^{-1}(4\cos^3(\theta) - 3\cos(\theta)) \]

2. **Use the identity for cosine of triple angle**:
The expression \( 4\cos^3(\theta) - 3\cos(\theta) \) can be simplified using the triple angle formula:
\[ 4\cos^3(\theta) - 3\cos(\theta) = \cos(3\theta) \]
Therefore, we have:
\[ \cos^{-1}(4\cos^3(\theta) - 3\cos(\theta)) = \cos^{-1}(\cos(3\theta)) = 3\theta \]

3. **Set the equation**:
Now we can equate the two sides:
\[ 3\theta = a + b \cos^{-1}(\cos(\theta)) \]
Since \( \cos^{-1}(\cos(\theta)) = \theta \) for \( \theta \in [0, \pi] \), we can rewrite the equation as:
\[ 3\theta = a + b\theta \]

4. **Rearranging the equation**:
Rearranging gives:
\[ (3 - b)\theta = a \]

5. **Identify \( a \) and \( b \)**:
To find \( a \) and \( b \), we need to analyze the limits of \( x \):
- For \( x = 1 \) (which corresponds to \( \theta = 0 \)):
\[ a = 0 \quad \text{and} \quad b = 3 \]
- For \( x = \frac{1}{2} \) (which corresponds to \( \theta = \frac{\pi}{3} \)):
\[ 3\left(\frac{\pi}{3}\right) = a + b\left(\frac{\pi}{3}\right) \]
This gives us the same values for \( a \) and \( b \).

6. **Calculate the limit**:
Now, we need to find:
\[ \lim_{y \to a} b \cos(y) \]
Substituting the values of \( a \) and \( b \):
\[ \lim_{y \to 0} 3 \cos(y) \]
As \( y \to 0 \):
\[ \cos(0) = 1 \]
Therefore:
\[ 3 \cdot 1 = 3 \]

### Final Answer:
The value of \( \lim_{y \to a} b \cos(y) \) is \( \boxed{3} \).