A function is defined by `f(x)=2+(x-1)^(2//3) on [0,2]`. Which of the following is not correct?

Options

A. f is not derivable in (0,2)
B. f is not continuous in [0,2]
C. `f(0)=f(2)`
D. Rolle's theorem is applicable on [0,2]

Explanation

To solve the problem step by step, we will analyze the function \( f(x) = 2 + (x - 1)^{2/3} \) on the interval \([0, 2]\) and determine which statement about the function is not correct.

### Step 1: Define the function
The function is given as:
\[ f(x) = 2 + (x - 1)^{2/3} \]

### Step 2: Check the continuity of the function
To check the continuity of \( f(x) \) on the interval \([0, 2]\), we need to evaluate the function at the endpoints and check if it is defined everywhere in the interval.

- **Evaluate \( f(0) \)**:
\[ f(0) = 2 + (0 - 1)^{2/3} = 2 + (-1)^{2/3} = 2 + 1 = 3 \]

- **Evaluate \( f(2) \)**:
\[ f(2) = 2 + (2 - 1)^{2/3} = 2 + (1)^{2/3} = 2 + 1 = 3 \]

Since \( f(0) = 3 \) and \( f(2) = 3 \), the function is continuous at the endpoints.

### Step 3: Check differentiability of the function
Next, we need to find the derivative \( f'(x) \) and check where it exists.

- **Differentiate \( f(x) \)**:
Using the power rule:
\[ f'(x) = 0 + \frac{2}{3}(x - 1)^{-1/3} \cdot (1) = \frac{2}{3}(x - 1)^{-1/3} \]

The derivative \( f'(x) \) is undefined when \( x - 1 = 0 \) (i.e., at \( x = 1 \)). Therefore, \( f'(x) \) does not exist at \( x = 1 \).

### Step 4: Analyze the implications of differentiability
Since \( f'(x) \) does not exist at \( x = 1 \), the function \( f(x) \) is not differentiable at that point.

### Step 5: Apply Rolle's Theorem
Rolle's Theorem states that if a function is continuous on a closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), then there exists at least one \( c \) in \((a, b)\) such that \( f'(c) = 0 \).

Since \( f(x) \) is not differentiable at \( x = 1 \), we cannot apply Rolle's Theorem on the interval \([0, 2]\).

### Conclusion
Now we can summarize our findings:
1. The function is continuous on \([0, 2]\).
2. The function is not differentiable at \( x = 1 \).
3. Since the function is not differentiable on the interval, Rolle's Theorem cannot be applied.

Thus, the statement that is not correct is the one that claims Rolle's Theorem is applicable.

### Final Answer
The option that is not correct is:
**Option 4: Rolle's Theorem is applicable.**