Rationale
(2, 4) must also lie on the graph of y = f(x).
Given that ( f'(-2) = f(2) ) due to the functional relationship ( f'(x) = f(-x) ), and knowing that ( f'(-2) = 4 ) (since the point (-2, 4) is on the graph), we conclude that ( f(2) = 4 ). Therefore, the point (2, 4) must also lie on the graph.
A) (-2, -4)
This point does not have to lie on the graph because there is no information suggesting that the function is odd or that values are negated at the corresponding points. The point (-2, 4) indicates that ( f(-2) = 4 ), not -4.
B) (0, 0)
This point cannot be concluded from the given information. The data provided does not imply that ( f(0) = 0 ) or that the function has any specific characteristics at zero. Hence, we cannot confirm this point lies on the graph.
C) (0, 4)
Similar to option B, there is no evidence in the function's definition or the given point to support that ( f(0) = 4 ). The relationship does not provide any information regarding the value of the function at x = 0.
D) (2, -4)
This point is incorrect because ( f(2) = 4 ) as derived from the function’s characteristics, and thus cannot equal -4. The value at x = 2 remains positive based on the information provided.
E) (2, 4)
This point must lie on the graph as it follows directly from the derivative condition and the known point (-2, 4). Since ( f'(-2) = 4 ) translates to ( f(2) = 4 ), this point is confirmed.
Conclusion
The function's properties dictate that if ( f'(-2) = f(2) ) and given the point (-2, 4), it follows that (2, 4) must also lie on the graph of ( y = f(x) ). The other options do not satisfy the conditions derived from the function's behavior and the given point, preserving the necessary relationship between the values of the function at symmetric points.
