Let `f(x)=sin^(-1){xsqrt(1-x)-sqrt(x(1-x^(2))}}, AA 0le xle1`
then `f(x)` is

Options

A. negative
B. positive
C. non - negative
D. non - positive

Answer

To solve the problem, we need to analyze the function:

\[ f(x) = \sin^{-1} \left( x \sqrt{1-x} - \sqrt{x(1-x^2)} \right) \]

for \( 0 \leq x \leq 1 \).

### Step 1: Simplifying the expression inside the inverse sine

We can rewrite the function \( f(x) \) in a more manageable form. Notice that:

\[ \sqrt{1-x^2} = \sqrt{(1-x)(1+x)} \]

This suggests that we can express \( f(x) \) in terms of sine functions.

### Step 2: Rewrite the expression

We can rewrite the function as follows:

\[ f(x) = \sin^{-1} \left( x \sqrt{1-x} - \sqrt{x(1-x^2)} \right) \]

Now, we simplify the expression inside the sine inverse:

1. The first term is \( x \sqrt{1-x} \).
2. The second term can be simplified as \( \sqrt{x(1-x^2)} = \sqrt{x(1-x)(1+x)} \).

### Step 3: Factor out common terms

We can factor out \( \sqrt{1-x} \):

\[ f(x) = \sin^{-1} \left( \sqrt{1-x} \left( x - \sqrt{x(1+x)} \right) \right) \]

### Step 4: Analyze the behavior of the function

To determine whether \( f(x) \) is positive, negative, or non-positive, we need to analyze the term \( x - \sqrt{x(1+x)} \):

1. For \( x = 0 \):
\[ f(0) = \sin^{-1}(0) = 0 \]

2. For \( x = 1 \):
\[ f(1) = \sin^{-1}(0) = 0 \]

3. For \( 0 < x < 1 \):
- We need to check if \( x - \sqrt{x(1+x)} \) is non-positive.

### Step 5: Check the inequality

To check if \( x - \sqrt{x(1+x)} \leq 0 \):

Squaring both sides (valid since both sides are non-negative for \( 0 \leq x \leq 1 \)):

\[ x^2 \leq x(1+x) \]

This simplifies to:

\[ x^2 \leq x + x^2 \]
\[ 0 \leq x \]

This is always true for \( x \geq 0 \).

### Conclusion

Since \( f(x) \) evaluates to 0 at both endpoints and is non-positive for \( 0 < x < 1 \), we conclude that:

\[ f(x) \text{ is non-positive for } 0 \leq x \leq 1. \]

Thus, the final answer is:

**f(x) is a non-positive function.**