Consider the following statements I Every function has a primitive II A primitive of a function is unique Which of the statements given above is/are correct?

Options

A. A) Only I
B. b) Only II
C. c) Both I nor 11
D. d) Neither I nor II

Answer

To solve the question regarding the correctness of the statements about primitives of functions, we will analyze each statement step by step.

### Step 1: Analyze Statement I
**Statement I:** "Every function has a primitive."

- A primitive of a function (also known as an antiderivative) is a function whose derivative is the original function.
- For a function to have a primitive, it must be continuous. However, not every function is continuous. For example, the function f(x) = 1/x is not defined at x = 0 and thus does not have a primitive over the entire real line.
- Therefore, this statement is **incorrect**.

### Step 2: Analyze Statement II
**Statement II:** "A primitive of a function is unique."

- The uniqueness of a primitive can be understood through the Fundamental Theorem of Calculus. If F(x) is a primitive of f(x), then any other primitive G(x) of f(x) can be expressed as G(x) = F(x) + C, where C is a constant.
- This means that while there can be many primitives for a function (differing by a constant), they all share the same form up to a constant. Hence, the statement about uniqueness refers to the fact that all primitives differ only by a constant.
- Therefore, this statement is **correct**.

### Conclusion
- Based on the analysis:
- Statement I is incorrect.
- Statement II is correct.

Thus, the correct answer is **Option B: 2**.

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