Final answer:
The correct answer is A) A curve that is strictly increasing. This is because a positive derivative indicates that the function is increasing for all values of x.
Step-by-step explanation:
If f'(x) > 0 for all real numbers x, it means that the function f is always increasing. This is because the derivative f'(x) represents the slope of the tangent line to the curve of the function f at any point x. A positive slope indicates that as x increases, f(x) also increases.
Given that knowledge, the curve that could be part of the graph f is A) A curve that is strictly increasing. Option B, C, and D do not fit the condition that f'(x) > 0. A strictly decreasing curve would imply f'(x) < 0. A curve with horizontal tangents would imply f'(x) = 0. And a curve that is symmetric about the y-axis does not necessarily relate to the sign of its derivative.