Question

Write the condition for the function \( f(x) \), to be strictly increasing, for all \( x \in \mathbb{R} \).

Hint:

For a function to be strictly increasing, its derivative must be positive over the entire domain.

Answer 1

Step 1: Recall the condition for strict increase.
A function \( f(x) \) is strictly increasing on an interval if its derivative is positive on that interval. That is, \[ f'(x) > 0 \text{for all } x \in \mathbb{R}. \]

Step 2: State the condition.
Thus, for \( f(x) \) to be strictly increasing for all \( x \in \mathbb{R} \), the condition is: \[ f'(x) > 0 \text{for all } x \in \mathbb{R}. \]

Final Answer: \[ \boxed{f'(x) > 0 \text{ for all } x \in \mathbb{R}} \]