A function f is defined by `f(x)=x^(2)" in "[2,3]`. Which of the following is correct ?

Options

A. Rolle's theorem is satisfied in [2,3]
B. Rolle's theorem is not satisfied in [2,3]
C. f is not continuous on [2,3]
D. f is not differentiable in (2,3)

Answer

To determine which statement regarding the function \( f(x) = x^2 \) defined on the interval \([2, 3]\) is correct, we will check the conditions of Rolle's Theorem.

### Step-by-Step Solution:

1. **Identify the function and interval**:
- The function is \( f(x) = x^2 \).
- The interval is \([2, 3]\).

2. **Check continuity**:
- A function is continuous on an interval if it is defined for all points in that interval and does not have any breaks, jumps, or asymptotes.
- Since \( f(x) = x^2 \) is a polynomial function, it is continuous everywhere, including on the interval \([2, 3]\).

3. **Check differentiability**:
- A function is differentiable on an interval if it has a derivative at all points in that interval.
- The derivative of \( f(x) = x^2 \) is \( f'(x) = 2x \), which is also a polynomial and thus differentiable everywhere, including on \([2, 3]\).

4. **Check the values at the endpoints**:
- Calculate \( f(a) \) and \( f(b) \) where \( a = 2 \) and \( b = 3 \):
- \( f(2) = 2^2 = 4 \)
- \( f(3) = 3^2 = 9 \)
- Now, check if \( f(a) = f(b) \):
- \( f(2) \neq f(3) \) since \( 4 \neq 9 \).

5. **Conclusion**:
- Since \( f(a) \) is not equal to \( f(b) \), the third condition of Rolle's Theorem is not satisfied. Therefore, Rolle's Theorem cannot be applied to this function on the interval \([2, 3]\).

### Final Answer:
- The correct statement is that Rolle's Theorem is not satisfied for the function \( f(x) = x^2 \) on the interval \([2, 3]\).