Let a function f(x) defined on [3,6] be given by `f(x)={{:(,log_(e)[x],3 le x lt5),(,|log_(e)x|,5 le x lt 6):}` then f(x) is

Options

A. continuous and differentiable on [3,6]
B. continuous on [3,6] but not differentiable at x=4,5
C. differentiable on [3,6] but not continuous at x=4,5
D. none of these

Explanation

To determine the properties of the function \( f(x) \) defined on the interval \([3, 6]\), we analyze it step by step.

### Step 1: Define the function
The function \( f(x) \) is defined as follows:
\[ f(x) = \begin{cases} \log_e(x) & \text{for } 3 \leq x < 5 \\ |\log_e(x)| & \text{for } 5 \leq x \leq 6 \end{cases} \]

### Step 2: Analyze continuity at critical points
We need to check the continuity of \( f(x) \) at the points where the definition of the function changes, specifically at \( x = 5 \) and \( x = 4 \).

#### At \( x = 4 \):
- **Left-hand limit**:
\[ \lim_{x \to 4^-} f(x) = \log_e(4) \]
- **Right-hand limit**:
\[ \lim_{x \to 4^+} f(x) = \log_e(4) \]
- Since both limits are equal, \( f(x) \) is continuous at \( x = 4 \).

#### At \( x = 5 \):
- **Left-hand limit**:
\[ \lim_{x \to 5^-} f(x) = \log_e(5) \]
- **Right-hand limit**:
\[ \lim_{x \to 5^+} f(x) = |\log_e(5)| = \log_e(5) \quad (\text{since } \log_e(5) > 0) \]
- Since both limits are equal, \( f(x) \) is continuous at \( x = 5 \).

### Step 3: Check differentiability at critical points
To determine differentiability, we need to check if the function is continuous first, which we have established.

#### At \( x = 4 \):
- The function is continuous, but we need to check the derivative:
\[ f'(x) = \frac{d}{dx}(\log_e(x)) = \frac{1}{x} \quad \text{for } 3 < x < 5 \]
- The derivative at \( x = 4 \) is:
\[ f'(4) = \frac{1}{4} \]

#### At \( x = 5 \):
- The left-hand derivative:
\[ \lim_{h \to 0} \frac{f(5-h) - f(5)}{-h} = \lim_{h \to 0} \frac{\log_e(5-h) - \log_e(5)}{-h} = \frac{1}{5} \]
- The right-hand derivative:
\[ \lim_{h \to 0} \frac{f(5+h) - f(5)}{h} = \lim_{h \to 0} \frac{|\log_e(5+h)| - \log_e(5)}{h} = \frac{1}{5} \]
- Since both derivatives are equal, \( f(x) \) is differentiable at \( x = 5 \).

### Conclusion
- The function \( f(x) \) is continuous on the interval \([3, 6]\) and differentiable on the interval \((3, 6)\) except at points \( x = 4 \) and \( x = 5 \) where it is continuous and differentiable.

### Final Answer
The correct option is: **Option 2: Continuous on \([3, 6]\) but not differentiable at \( x = 4 \) and \( x = 5\)**.