Rationale
(2, 4) must also lie on the graph of y = f(x).
Given that f'(x) = f(-x), we can infer a symmetry in the function f(x). Since the point (-2, 4) indicates that f(-2) = 4, we also find that f(2) must equal 4 by the property of the function, leading us to conclude that the point (2, 4) lies on the graph.
A) (-2,-4)
This point does not have to lie on the graph since it contradicts the information given. The point (-2, 4) indicates that f(-2) = 4, so there is no basis for asserting f(-2) = -4.
B) (0,0)
While (0,0) could be a potential point on some functions, there is no information given that necessitates it to be on the graph. The symmetry condition and the specific point (-2, 4) do not imply that f(0) must equal 0.
C) (0,4)
Similar to option B, there is no evidence that f(0) must equal 4 based on the information provided. The function's behavior at x = 0 is not determined by the property or the given point.
D) (2,-4)
This point cannot be on the graph since we established that f(2) = 4 based on the symmetry derived from f'(x) = f(-x). Therefore, there is no support for f(2) being -4.
E) (2,4)
Since f(-2) = 4 indicates that f(2) must also equal 4, the point (2, 4) is confirmed to lie on the graph of y = f(x).
Conclusion
The unique property of the function f, defined by f'(x) = f(-x), implies that for every point on the graph, a corresponding symmetric point must also exist. Given the point (-2, 4), we deduced that (2, 4) is a required point on the graph. Other options either contradict the function's symmetry or do not follow logically from the given information.
