Consider the following statements :
1. Derivative of f(x) may not exist at some point.
2. Derivative of f(x) may exist finitely at some point.
3. Derivative of f(x) may be infinite (geometircally) at some point.
Which of the above statements are correct?

Options

A. 1 and 2 olny
B. 2 and 3 only
C. 1 and 3 only
D. 1, 2 and 3

Explanation

To determine the correctness of the given statements regarding derivatives, we will analyze each statement one by one.

### Step 1: Analyze Statement 1
**Statement 1:** Derivative of f(x) may not exist at some point.

**Explanation:** This statement is true. There are functions where the derivative does not exist at certain points. A classic example is the function \( f(x) = |x| \). At \( x = 0 \), the function has a sharp corner, and thus the derivative is not defined at that point.

### Step 2: Analyze Statement 2
**Statement 2:** Derivative of f(x) may exist finitely at some point.

**Explanation:** This statement is also true. There are many functions whose derivatives exist and are finite at certain points. For example, the function \( f(x) = x^2 \) has a derivative \( f'(x) = 2x \), which is finite for all \( x \). Thus, the derivative can exist finitely at some points.

### Step 3: Analyze Statement 3
**Statement 3:** Derivative of f(x) may be infinite (geometrically) at some point.

**Explanation:** This statement is true as well. A derivative can be infinite at certain points, which typically occurs at vertical tangents. For example, consider the function \( f(x) = x^{1/3} \). At \( x = 0 \), the derivative \( f'(x) = \frac{1}{3} x^{-2/3} \) approaches infinity, indicating a vertical tangent.

### Conclusion
All three statements are correct. Therefore, the correct option is that all statements (1, 2, and 3) are true.

### Final Answer
**Correct Statements:** 1, 2, and 3 are correct.

---