Let `f (x)= {{:(x [x] , 0 le x lt 2),( (x-1), 2 le x le 3):}` where [x]= greatest integer less than or equal to x, then:
The number of values of x for `x in [0,3]` where `f (x)` is non-differentiable is :

Options

A. 0
B. 1
C. 2
D. 3

Answer

To determine the number of values of \( x \) in the interval \([0, 3]\) where the function \( f(x) \) is non-differentiable, we will analyze the function piecewise.

The function is defined as:
\[ f(x) = \begin{cases} x \cdot [x] & \text{for } 0 \leq x < 2 \\ x - 1 & \text{for } 2 \leq x \leq 3 \end{cases} \]
where \([x]\) denotes the greatest integer less than or equal to \( x \).

### Step 1: Analyze the first piece \( f(x) = x \cdot [x] \) for \( 0 \leq x < 2 \)

1. For \( 0 \leq x < 1 \):
- Here, \([x] = 0\), so \( f(x) = x \cdot 0 = 0 \).

2. For \( 1 \leq x < 2 \):
- Here, \([x] = 1\), so \( f(x) = x \cdot 1 = x \).

### Step 2: Identify points of non-differentiability in the first piece

- At \( x = 1 \):
- The function changes from \( f(x) = 0 \) to \( f(x) = x \).
- The left-hand limit \( f(1^-) = 0 \) and the right-hand limit \( f(1^+) = 1 \).
- Since these two limits are not equal, \( f(x) \) is non-differentiable at \( x = 1 \).

### Step 3: Analyze the second piece \( f(x) = x - 1 \) for \( 2 \leq x \leq 3 \)

- This is a linear function, and linear functions are differentiable everywhere. Thus, there are no points of non-differentiability in this interval.

### Step 4: Identify points of non-differentiability at the boundary between pieces

- At \( x = 2 \):
- From the first piece, as \( x \) approaches 2 from the left, \( f(2^-) = 2 \cdot [2] = 2 \).
- From the second piece, \( f(2^+) = 2 - 1 = 1 \).
- Since these two limits are not equal, \( f(x) \) is non-differentiable at \( x = 2 \).

### Conclusion

The function \( f(x) \) is non-differentiable at two points: \( x = 1 \) and \( x = 2 \).

Thus, the number of values of \( x \) for \( x \in [0, 3] \) where \( f(x) \) is non-differentiable is:
\[ \boxed{2} \]