If the function `f(x)=cos^(-1)(x^((3)/(2))-sqrt(1-x-x^(2)+x^(3))) (" where, "AA 0 lt x lt 1),` then the value of `|sqrt3f'((1)/(2))|` is equal to `("take "sqrt3=1.73)`

Answer

To solve the problem, we need to find the value of \( | \sqrt{3} f'(\frac{1}{2}) | \) for the function \( f(x) = \cos^{-1} \left( x^{\frac{3}{2}} - \sqrt{1 - x - x^2 + x^3} \right) \) where \( 0 < x < 1 \).

### Step 1: Simplify the function \( f(x) \)

Given:
\[ f(x) = \cos^{-1} \left( x^{\frac{3}{2}} - \sqrt{1 - x - x^2 + x^3} \right) \]

First, we simplify the expression inside the cosine inverse. We can rewrite \( x^{\frac{3}{2}} \) as \( x \sqrt{x} \).

Next, we simplify \( \sqrt{1 - x - x^2 + x^3} \):
\[ 1 - x - x^2 + x^3 = (1 - x)(1 + x + x^2) \]
Thus,
\[ \sqrt{1 - x - x^2 + x^3} = \sqrt{(1 - x)(1 + x + x^2)} \]

Now, we can express \( f(x) \) as:
\[ f(x) = \cos^{-1} \left( x \sqrt{x} - \sqrt{(1 - x)(1 + x + x^2)} \right) \]

### Step 2: Use the identity for cosine inverse

Using the identity:
\[ \cos^{-1}(A) + \cos^{-1}(B) = \cos^{-1}(AB - \sqrt{(1 - A^2)(1 - B^2)}) \]
we can express \( f(x) \) as:
\[ f(x) = \cos^{-1}(x) + \cos^{-1}(\sqrt{x}) \]

### Step 3: Differentiate \( f(x) \)

Now we differentiate \( f(x) \):
\[ f'(x) = -\frac{1}{\sqrt{1 - x^2}} - \frac{1}{\sqrt{1 - x}} \cdot \frac{1}{2\sqrt{x}} \]

This simplifies to:
\[ f'(x) = -\frac{1}{\sqrt{1 - x^2}} - \frac{1}{2\sqrt{x(1 - x)}} \]

### Step 4: Evaluate \( f'(\frac{1}{2}) \)

Now we substitute \( x = \frac{1}{2} \):
\[ f'(\frac{1}{2}) = -\frac{1}{\sqrt{1 - \left(\frac{1}{2}\right)^2}} - \frac{1}{2\sqrt{\frac{1}{2}(1 - \frac{1}{2})}} \]
Calculating each term:
\[ -\frac{1}{\sqrt{1 - \frac{1}{4}}} = -\frac{1}{\sqrt{\frac{3}{4}}} = -\frac{2}{\sqrt{3}} \]
And for the second term:
\[ -\frac{1}{2\sqrt{\frac{1}{2} \cdot \frac{1}{2}}} = -\frac{1}{2 \cdot \frac{1}{2}} = -1 \]
Thus,
\[ f'(\frac{1}{2}) = -\frac{2}{\sqrt{3}} - 1 \]

### Step 5: Calculate \( | \sqrt{3} f'(\frac{1}{2}) | \)

Now we calculate:
\[ | \sqrt{3} f'(\frac{1}{2}) | = | \sqrt{3} \left( -\frac{2}{\sqrt{3}} - 1 \right) | \]
This simplifies to:
\[ | -2 - \sqrt{3} | = 2 + \sqrt{3} \]

Substituting \( \sqrt{3} = 1.73 \):
\[ 2 + 1.73 = 3.73 \]

### Final Answer

Thus, the value of \( | \sqrt{3} f'(\frac{1}{2}) | \) is:
\[ \boxed{3.73} \]