If `cos^(-1)(4x^3-3x)=a+bcos^(-1)x" for "-1 lt x lt -1/2,` then [a + b+ 4] is equal to {where [] denotes G.I.F}`

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) in the equation:

\[ \cos^{-1}(4x^3 - 3x) = a + b \cos^{-1}(x) \]

for \( -1 < x < -\frac{1}{2} \).

### Step 1: Identify the Range of \( x \)
We know that \( x \) lies in the interval \( -1 < x < -\frac{1}{2} \).

### Step 2: Use the Identity for \( \cos^{-1} \)
We can use the trigonometric identity that relates \( \cos^{-1} \) of a polynomial in \( x \) to \( \cos^{-1}(x) \):
\[ \cos^{-1}(4x^3 - 3x) = 3 \cos^{-1}(x) \]
This identity holds when \( x \) is in the range \( -1 \leq x \leq 1 \).

### Step 3: Determine the Appropriate Form
Since we are looking for the expression in the form \( a + b \cos^{-1}(x) \), we can rewrite the equation:
\[ \cos^{-1}(4x^3 - 3x) = 3 \cos^{-1}(x) \]

### Step 4: Compare the Forms
Now, we can compare:
\[ 3 \cos^{-1}(x) = a + b \cos^{-1}(x) \]
From this, we can see that:
- \( b = 3 \)
- \( a = 0 \)

### Step 5: Calculate \( a + b + 4 \)
Now, we can calculate:
\[ a + b + 4 = 0 + 3 + 4 = 7 \]

### Step 6: Find the Greatest Integer Function
The greatest integer function \( [x] \) gives the largest integer less than or equal to \( x \). Thus:
\[ [a + b + 4] = [7] = 7 \]

### Final Answer
The final answer is:
\[ \boxed{7} \]