If `f(x)=cos^(-1)(x^((3)/(2))-sqrt(1-x-x^(2)+x^(3))),AA 0 le x le 1` then the minimum value of `f(x)` is

Answer

To find the minimum value of the function \( f(x) = \cos^{-1}\left(x^{\frac{3}{2}} - \sqrt{1 - x - x^2 + x^3}\right) \) for \( 0 \leq x \leq 1 \), we will follow these steps:

### Step 1: Simplify the expression inside the inverse cosine
We start with the expression inside the cosine inverse:

\[ x^{\frac{3}{2}} - \sqrt{1 - x - x^2 + x^3} \]

First, we simplify \( 1 - x - x^2 + x^3 \):

\[ 1 - x - x^2 + x^3 = 1 - x(1 + x - x^2) = 1 - x(1 + x - x^2) \]

### Step 2: Rewrite the function
Now, we can rewrite the function as:

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

### Step 3: Evaluate the function at the endpoints
Next, we evaluate \( f(x) \) at the endpoints of the interval \( [0, 1] \).

1. **At \( x = 0 \)**:
\[ f(0) = \cos^{-1}\left(0^{\frac{3}{2}} - \sqrt{1 - 0 - 0^2 + 0^3}\right) = \cos^{-1}(0 - 1) = \cos^{-1}(-1) = \pi \]

2. **At \( x = 1 \)**:
\[ f(1) = \cos^{-1}\left(1^{\frac{3}{2}} - \sqrt{1 - 1 - 1^2 + 1^3}\right) = \cos^{-1}(1 - 0) = \cos^{-1}(1) = 0 \]

### Step 4: Check for critical points
To find critical points, we need to differentiate \( f(x) \) and set the derivative to zero. However, since we are looking for minimum values, we can compare the values at the endpoints and check if there are any critical points in the interval.

### Step 5: Compare values
From our evaluations:
- \( f(0) = \pi \)
- \( f(1) = 0 \)

Since \( f(1) = 0 \) is less than \( f(0) = \pi \), we conclude that the minimum value of \( f(x) \) on the interval \( [0, 1] \) is:

\[ \text{Minimum value of } f(x) = 0 \]

### Final Answer
The minimum value of \( f(x) \) is \( 0 \).

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