Let f(x)`= [sin ^(4)x]` then ( where [.] represents the greatest integer function ).

Options

A. (a) f(x) is continuous at x=0
B. (b) f(x) is differentiable at x=0
C. (c) f(x) is non-differnentiable at x=0
D. (d) `f'(0) =1`

Answer

To solve the problem, we need to analyze the function \( f(x) = \lfloor \sin^4 x \rfloor \), where \( \lfloor . \rfloor \) represents the greatest integer function.

### Step-by-Step Solution:

1. **Understanding the Function**:
The function \( f(x) = \lfloor \sin^4 x \rfloor \) takes the value of \( \sin^4 x \) and applies the greatest integer function to it. This means we need to determine the range of \( \sin^4 x \).

2. **Finding the Range of \( \sin^4 x \)**:
The sine function, \( \sin x \), oscillates between -1 and 1. Therefore, \( \sin^4 x \) will oscillate between \( 0 \) and \( 1 \) because:
\[ 0 \leq \sin^2 x \leq 1 \implies 0 \leq \sin^4 x \leq 1. \]
This means \( \sin^4 x \) will take values from \( 0 \) to \( 1 \).

3. **Applying the Greatest Integer Function**:
Since \( \sin^4 x \) can take any value in the interval \( [0, 1) \) (it never actually reaches 1), the greatest integer function \( \lfloor \sin^4 x \rfloor \) will be:
\[ f(x) = \lfloor \sin^4 x \rfloor = 0 \quad \text{for all } x \text{ such that } \sin^4 x < 1. \]
Therefore, for all \( x \) in the interval \( (-\frac{\pi}{2}, \frac{\pi}{2}) \), \( f(x) = 0 \).

4. **Continuity and Differentiability**:
Since \( f(x) = 0 \) for all \( x \) in the interval \( (-\frac{\pi}{2}, \frac{\pi}{2}) \), it is a constant function. A constant function is continuous and differentiable everywhere. Therefore:
- \( f(x) \) is continuous at \( x = 0 \).
- \( f(x) \) is differentiable at \( x = 0 \).

5. **Finding the Derivative at \( x = 0 \)**:
The derivative of a constant function is zero. Thus:
\[ f'(0) = 0. \]

### Conclusion:
- The function \( f(x) = \lfloor \sin^4 x \rfloor \) is continuous and differentiable at \( x = 0 \).
- The value of \( f'(0) \) is \( 0 \).

### Summary of Results:
- \( f(x) \) is continuous at \( x = 0 \).
- \( f(x) \) is differentiable at \( x = 0 \).
- \( f'(0) = 0 \).