If `y = sin^-1(3x - 4x^3)`, then the number of points in `[-1, 1]`, where y is not differentiable is

Options

A. `f'(x)=-(3)/(sqrt(1-x^(2)))"for all "x in (-1,1)`
B. `f'(x)=(3)/(sqrt(1-x^(2)))"for all "x in [-1,1]`
C. `f'(x)={{:(,(3)/(sqrt(1-x^(2))),"if "-(1)/(2) lt x lt (1)/(2)),(,(-3)/(sqrt(1-x^(2))),"if "(1)/(2) lt x lt 1 or"," -1 lt x lt -(1)/(2)):}`
D. `f'(x)={{:(,(3)/(sqrt(1-x^(2))),"if "|x| lt (sqrt3)/(2)),(,(-3)/(sqrt(1-x^(2))),"if "1gt |x| gt (sqrt3)/(2)):}`

Explanation

To determine the number of points in the interval \([-1, 1]\) where the function \(y = \sin^{-1}(3x - 4x^3)\) is not differentiable, we need to analyze the conditions under which the derivative of \(y\) is defined.

### Step-by-step Solution:

1. **Identify the Function**:
We have the function:
\[ y = \sin^{-1}(3x - 4x^3) \]

2. **Differentiate the Function**:
We need to find the derivative \( \frac{dy}{dx} \). The derivative of \( \sin^{-1}(u) \) is given by:
\[ \frac{dy}{dx} = \frac{1}{\sqrt{1 - u^2}} \cdot \frac{du}{dx} \]
where \( u = 3x - 4x^3 \).

3. **Calculate \( \frac{du}{dx} \)**:
Differentiate \( u \):
\[ \frac{du}{dx} = 3 - 12x^2 \]

4. **Substitute into the Derivative**:
Now substituting \( u \) and \( \frac{du}{dx} \) into the derivative formula:
\[ \frac{dy}{dx} = \frac{1}{\sqrt{1 - (3x - 4x^3)^2}} \cdot (3 - 12x^2) \]

5. **Determine Where the Derivative is Undefined**:
The derivative \( \frac{dy}{dx} \) is undefined where the denominator is zero:
\[ 1 - (3x - 4x^3)^2 = 0 \]
This implies:
\[ (3x - 4x^3)^2 = 1 \]
Taking square roots gives:
\[ 3x - 4x^3 = 1 \quad \text{or} \quad 3x - 4x^3 = -1 \]

6. **Solve the Equations**:
- For \( 3x - 4x^3 = 1 \):
\[ 4x^3 - 3x + 1 = 0 \]
- For \( 3x - 4x^3 = -1 \):
\[ 4x^3 - 3x - 1 = 0 \]

7. **Finding Roots**:
We need to find the roots of these cubic equations within the interval \([-1, 1]\).

- For \( 4x^3 - 3x + 1 = 0 \):
- Testing \( x = 1 \):
\[ 4(1)^3 - 3(1) + 1 = 4 - 3 + 1 = 2 \quad (\text{not a root}) \]
- Testing \( x = -1 \):
\[ 4(-1)^3 - 3(-1) + 1 = -4 + 3 + 1 = 0 \quad (\text{is a root}) \]
- The other roots can be found using synthetic division or numerical methods.

- For \( 4x^3 - 3x - 1 = 0 \):
- Testing \( x = 1 \):
\[ 4(1)^3 - 3(1) - 1 = 4 - 3 - 1 = 0 \quad (\text{is a root}) \]
- The other roots can also be found similarly.

8. **Count the Points**:
After solving both cubic equations, we find that there are two points where the derivative is undefined in the interval \([-1, 1]\).

### Conclusion:
The total number of points in the interval \([-1, 1]\) where \(y\) is not differentiable is **2**.