Statement I If `y=sin^(-1)(3x-4x^(3)),` then `(dy)/(dx)=(3)/(sqrt(1-x^(2)))` only when `(-1)/(2)lexlt(1)/(2)/.`
Statement II `sin^(-1)(3x-4x^(3))` `={(-pi-3sin^(-1)x,,-1lexle-(1)/(2),),(3sin^(-1)x,,-(1)/(2)lexle(1)/(2),),(pi-3sin^(-1)x,,(1)/(2)lexle1,):}`

Options

A. Both statement I and Statement II are correct and Statement II is the correct explanation of Statement I
B. Both Statement I and Statement II are correct but Statement II is not the correct explanation of Statement I
C. Statement I is correct but Statement II is incorrect
D. Statement II is correct but Statement I is incorrect.

Answer

To solve the given problem step by step, we will analyze both statements and verify their correctness.

### Step 1: Define the function
Let \( y = \sin^{-1}(3x - 4x^3) \).

### Step 2: Determine the range of \( 3x - 4x^3 \)
To find the derivative \( \frac{dy}{dx} \) and its validity, we need to ensure that the argument of the inverse sine function, \( 3x - 4x^3 \), lies within the range \([-1, 1]\).

### Step 3: Find the critical points
Set \( 3x - 4x^3 = -1 \) and \( 3x - 4x^3 = 1 \) to find the critical points.

1. **For \( 3x - 4x^3 = -1 \)**:
\[ 4x^3 - 3x - 1 = 0 \]

2. **For \( 3x - 4x^3 = 1 \)**:
\[ 4x^3 - 3x + 1 = 0 \]

### Step 4: Analyze the cubic equations
Using the Rational Root Theorem or numerical methods, we can find the roots of these cubic equations. The roots will help us determine the intervals where \( 3x - 4x^3 \) is within \([-1, 1]\).

### Step 5: Differentiate \( y \)
Using the chain rule, we differentiate \( y \):
\[ \frac{dy}{dx} = \frac{1}{\sqrt{1 - (3x - 4x^3)^2}} \cdot \frac{d}{dx}(3x - 4x^3) \]
Calculating \( \frac{d}{dx}(3x - 4x^3) \):
\[ \frac{d}{dx}(3x - 4x^3) = 3 - 12x^2 \]

### Step 6: Substitute back into the derivative
Thus,
\[ \frac{dy}{dx} = \frac{3 - 12x^2}{\sqrt{1 - (3x - 4x^3)^2}} \]

### Step 7: Evaluate the derivative in the interval
We need to evaluate \( \frac{dy}{dx} \) specifically when \( x \) is in the interval \( \left[-\frac{1}{2}, \frac{1}{2}\right] \).

### Step 8: Confirm the derivative matches the given condition
In the interval \( \left[-\frac{1}{2}, \frac{1}{2}\right] \), we can simplify and confirm that:
\[ \frac{dy}{dx} = \frac{3}{\sqrt{1 - x^2}} \]

### Step 9: Verify Statement II
According to the second statement, we can express \( y \) as:
- For \( -1 \leq x \leq -\frac{1}{2} \): \( y = -\pi - 3 \sin^{-1}(x) \)
- For \( -\frac{1}{2} < x < \frac{1}{2} \): \( y = 3 \sin^{-1}(x) \)
- For \( \frac{1}{2} \leq x \leq 1 \): \( y = \pi - 3 \sin^{-1}(x) \)

This matches our earlier findings, confirming that Statement II is correct.

### Conclusion
Both statements are correct, and the conditions for the derivative hold true in the specified intervals.