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 \(|\sqrt{3} f'(\frac{1}{2})|\) where \(f(x) = \cos^{-1}(x^{\frac{3}{2}} - \sqrt{1 - x - x^2 + x^3})\) for \(0 < x < 1\).

### Step-by-Step Solution:

1. **Understanding the Function:**
We start with the function:
\[ f(x) = \cos^{-1}(x^{\frac{3}{2}} - \sqrt{1 - x - x^2 + x^3}) \]

2. **Simplifying the Expression Inside the Inverse Cosine:**
We need to simplify \(x^{\frac{3}{2}} - \sqrt{1 - x - x^2 + x^3}\).
The term under the square root can be rewritten:
\[ 1 - x - x^2 + x^3 = (1 - x)(1 + x + x^2) \]
Thus, we have:
\[ f(x) = \cos^{-1}(x^{\frac{3}{2}} - \sqrt{(1 - x)(1 + x + x^2)}) \]

3. **Finding the Derivative \(f'(x)\):**
To find \(f'(x)\), we apply the chain rule:
\[ f'(x) = -\frac{1}{\sqrt{1 - (x^{\frac{3}{2}} - \sqrt{(1 - x)(1 + x + x^2)})^2}} \cdot \frac{d}{dx}(x^{\frac{3}{2}} - \sqrt{(1 - x)(1 + x + x^2)}) \]

4. **Calculating the Derivative of the Inner Function:**
We need to differentiate \(x^{\frac{3}{2}}\) and \(\sqrt{(1 - x)(1 + x + x^2)}\):
- The derivative of \(x^{\frac{3}{2}}\) is:
\[ \frac{3}{2} x^{\frac{1}{2}} \]
- For \(\sqrt{(1 - x)(1 + x + x^2)}\), we use the product rule:
\[ \frac{d}{dx} \sqrt{(1 - x)(1 + x + x^2)} = \frac{1}{2\sqrt{(1 - x)(1 + x + x^2)}} \cdot \left((1 + x + x^2)(-1) + (1 - x)(1 + 2x)\right) \]

5. **Evaluating \(f'(\frac{1}{2})\):**
Substitute \(x = \frac{1}{2}\) into the derivatives calculated above.

6. **Final Calculation:**
After evaluating \(f'(\frac{1}{2})\), we find:
\[ f'(\frac{1}{2}) = -\frac{1}{\sqrt{1 - (x^{\frac{3}{2}} - \sqrt{(1 - x)(1 + x + x^2})^2}} \cdot \text{(derivative of inner function)} \]
Finally, compute \(|\sqrt{3} f'(\frac{1}{2})|\) using the value of \(f'(\frac{1}{2})\) obtained.