which of the following is true for `x in [0,1]`?

Options

A. `sin^(-1)x+x^(2)-x(9-x^(2))/(3)le0`
B. `sin^(-1)x+x^(2)-x(9-x^(2))/(3)ge0`
C. `sin^(-1)x+x^(2)-x(9-x^(2))/(3)le0`
D. `sin^(-1)x+x^(2)-x(9-x^(2))/(3)ge0`

Explanation

To solve the problem, we need to analyze the function given in the question and determine its behavior over the interval \( x \in [0, 1] \). The function we are considering is:

\[ f(x) = \sin^{-1}(x) + \frac{x^3}{3} - 3x + x^2 \]

### Step 1: Differentiate the Function
First, we differentiate \( f(x) \) with respect to \( x \):

\[ f'(x) = \frac{d}{dx}(\sin^{-1}(x)) + \frac{d}{dx}\left(\frac{x^3}{3}\right) - \frac{d}{dx}(3x) + \frac{d}{dx}(x^2) \]

Calculating each derivative:

- The derivative of \( \sin^{-1}(x) \) is \( \frac{1}{\sqrt{1 - x^2}} \).
- The derivative of \( \frac{x^3}{3} \) is \( x^2 \).
- The derivative of \( -3x \) is \( -3 \).
- The derivative of \( x^2 \) is \( 2x \).

Combining these, we get:

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

### Step 2: Analyze the Derivative
Next, we need to analyze the sign of \( f'(x) \) over the interval \( [0, 1] \):

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

### Step 3: Find Critical Points
Set \( f'(x) = 0 \) to find critical points:

\[ \frac{1}{\sqrt{1 - x^2}} + x^2 + 2x - 3 = 0 \]

This equation can be complex to solve directly, but we can analyze the behavior of \( f'(x) \) at the endpoints of the interval.

### Step 4: Evaluate at Endpoints
Calculate \( f(0) \) and \( f(1) \):

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

2. **At \( x = 1 \)**:
\[ f(1) = \sin^{-1}(1) + \frac{1^3}{3} - 3(1) + 1^2 = \frac{\pi}{2} + \frac{1}{3} - 3 + 1 = \frac{\pi}{2} - \frac{8}{3} \]

### Step 5: Determine Global Maxima and Minima
Now we need to determine if \( f(x) \) has a global maximum or minimum in the interval \( [0, 1] \):

- Since \( f(0) = 0 \) and \( f(1) < 0 \) (as \( \frac{\pi}{2} - \frac{8}{3} < 0 \)), we can conclude that \( f(x) \) has a global maximum at \( x = 0 \).

### Conclusion
From our analysis, we can conclude that:

\[ f(x) \leq 0 \text{ for } x \in [0, 1] \]

Thus, the correct statement is that \( f(x) \) is less than or equal to 0 for \( x \in [0, 1] \).