`f(x)` is a strictly increasing function, if `f'(x)` is :

Options

A. positive
B. negative
C. 0
D. None of these.

Answer

To solve the problem, we need to analyze the given information about the function \( f(x) \) and its derivative \( f'(x) \).

### Step-by-Step Solution:

1. **Understanding Strictly Increasing Function**:
A function \( f(x) \) is said to be strictly increasing if for any two points \( x_1 \) and \( x_2 \) in its domain, where \( x_1 < x_2 \), it holds that \( f(x_1) < f(x_2) \).

2. **Derivative and Increasing Functions**:
The derivative \( f'(x) \) of a function gives us the slope of the tangent line to the curve at any point \( x \). For a function to be strictly increasing, the slope of the tangent line must be positive.

3. **Condition for Strictly Increasing**:
Therefore, the condition for \( f(x) \) to be strictly increasing is:
\[ f'(x) > 0 \quad \text{for all } x \text{ in the domain of } f. \]

4. **Conclusion**:
Since \( f'(x) \) must be greater than zero for \( f(x) \) to be strictly increasing, we conclude that:
\[ f'(x) \text{ is positive.} \]

### Final Answer:
The correct option is that \( f'(x) \) is **positive**.

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