Statement I For the function
`f(x)={:{(15-x,xlt2),(2x-3,xge2):}x=2` has neither a maximum nor a minimum point.
Statament II ff'(x) does not exist at x=2.

Options

A. Statement I is true, Statement II is also true, Statement II is the correct explanation of statement I.
B. Statement I is true, Statement II is also true, Statement II is not the correct explanation of Statement I.
C. Statement I is true, Statement II is false
D. Statement I is false, Statement II is true

Answer

To solve the problem, we need to analyze the given piecewise function and determine the validity of the statements regarding its maximum and minimum points, as well as the existence of the derivative at a specific point.

### Given Function:
The function is defined as:
\[ f(x) = \begin{cases} 15 - x & \text{for } x < 2 \\ 2x - 3 & \text{for } x \geq 2 \end{cases} \]

### Step 1: Evaluate the function at \( x = 2 \)
To find the value of the function at \( x = 2 \):
\[ f(2) = 2(2) - 3 = 4 - 3 = 1 \]

### Step 2: Analyze the behavior of the function around \( x = 2 \)
For \( x < 2 \):
\[ f(x) = 15 - x \]
This is a decreasing linear function.

For \( x \geq 2 \):
\[ f(x) = 2x - 3 \]
This is an increasing linear function.

### Step 3: Determine the left-hand limit and right-hand limit at \( x = 2 \)
- Left-hand limit as \( x \) approaches 2:
\[ \lim_{x \to 2^-} f(x) = 15 - 2 = 13 \]

- Right-hand limit as \( x \) approaches 2:
\[ \lim_{x \to 2^+} f(x) = 2(2) - 3 = 1 \]

### Step 4: Check for continuity at \( x = 2 \)
Since the left-hand limit (13) does not equal the right-hand limit (1), the function is not continuous at \( x = 2 \). Therefore, it cannot have a maximum or minimum at this point.

### Step 5: Analyze the derivative at \( x = 2 \)
- For \( x < 2 \):
\[ f'(x) = -1 \]
- For \( x \geq 2 \):
\[ f'(x) = 2 \]

At \( x = 2 \), the left-hand derivative is -1 and the right-hand derivative is 2. Since these two values are not equal, \( f'(2) \) does not exist.

### Conclusion:
- **Statement I**: The function has neither a maximum nor a minimum point at \( x = 2 \) is **true**.
- **Statement II**: \( f'(x) \) does not exist at \( x = 2 \) is also **true**.

Thus, both statements are correct.

### Final Answer:
Both Statement I and Statement II are true.