16. Let f be a function such that f'(x)=f(-x) for all real numbers x. If the point (-2, 4) lies on the graph of y=f(x) in the xy-plane, which of the following points must also lie on the graph of y = f(x)?
Answer: E
The point (2, 4) must also lie on the graph of y = f(x).
Since f'(-2) = f(2) due to the property of the function f where f'(x) = f(-x), and given that f'(-2) is the derivative at the point where (-2, 4) lies, it follows that f(2) must equal 4.
A) (-2,-4)
This option is incorrect because it suggests that the function takes the value -4 at x = -2. Since the function value at (-2, 4) is positive, there is no justification for f(-2) = -4.
B) (0,0)
This option is incorrect because there is no information provided in the context that indicates f(0) must equal 0. The derivative condition does not imply anything about the value of the function at zero.
C) (0,4)
This option is incorrect as well. Similar to option B, no information leads to the conclusion that f(0) must equal 4, hence it cannot be determined from the provided conditions.
D) (2,-4)
This option is incorrect since it implies that the function value at x = 2 is -4. However, since we established that f(2) = 4, this contradicts the conclusion derived from the derivative condition.
E) (2,4)
This option is correct because, based on the derivative relationship f'(x) = f(-x) and the given point (-2, 4), we derive that f(2) must also equal 4, confirming that (2, 4) lies on the graph.
Conclusion
The correct answer is (2, 4) as it directly follows from the derivative relationship provided in the question. All other options fail because they either misinterpret the function's values or do not comply with the established properties of the function f. Thus, (2, 4) is the only point that must lie on the graph of y = f(x).