If `f(x)={{:(,x[x], 0 le x lt 2),(,(x-1)[x], 2 le x lt 3):}` where [.] denotes the greatest integer function, then

Options

A. both f'(1) and f'(2) do not exist
B. f'(1) exist but f'(2) does not exist
C. f'(2) exist but f'(1) does not exist
D. both f'(1) and f'(2) exist

Explanation

To solve the problem, we need to analyze the piecewise function given by:

\[ f(x) = \begin{cases} 0 & \text{for } 0 \leq x < 2 \\ x - 1 & \text{for } 2 \leq x < 3 \end{cases} \]

### Step 1: Determine the values of \( f(x) \) in the specified intervals.

- For \( 0 \leq x < 2 \):
- Since \( f(x) = 0 \), we have \( f(x) = 0 \) for all \( x \) in this interval.

- For \( 2 \leq x < 3 \):
- Here, \( f(x) = x - 1 \).

### Step 2: Check the continuity of \( f(x) \) at the boundaries.

- At \( x = 2 \):
- \( f(2) = 2 - 1 = 1 \) (from the second piece).
- The limit from the left as \( x \) approaches 2 is \( f(2^-) = 0 \).
- The limit from the right as \( x \) approaches 2 is \( f(2^+) = 1 \).

Since \( f(2^-) \neq f(2^+) \), \( f(x) \) is discontinuous at \( x = 2 \).

### Step 3: Check the differentiability of \( f(x) \).

- For \( 0 \leq x < 2 \):
- The function is constant (0), so \( f'(x) = 0 \).

- For \( 2 < x < 3 \):
- The derivative \( f'(x) = 1 \) (since the derivative of \( x - 1 \) is 1).

### Step 4: Analyze the points of interest.

- At \( x = 1 \):
- \( f'(1) = 0 \) (from the left interval).

- At \( x = 2 \):
- The left-hand derivative \( f'(2^-) = 0 \) and the right-hand derivative \( f'(2^+) = 1 \).

Since \( f'(2^-) \neq f'(2^+) \), \( f'(2) \) does not exist.

### Conclusion:

The function \( f(x) \) is discontinuous at \( x = 2 \) and non-differentiable at both \( x = 1 \) and \( x = 2 \).

Thus, the correct answer is that \( f'(1) \) and \( f'(2) \) do not exist.