Final answer:
The point of maximum curvature for the curve y = 4e^x is where the second derivative is the highest. The second derivative, y'' = 4e^x, increases with x; therefore, the point of maximum curvature is at the highest given value of x, which is x = 2.
Step-by-step explanation:
To find the point of maximum curvature for the curve given by the equation y = 4ex, we must calculate the second derivative of the function and analyze it to find the point where the curvature is greatest. The curvature of a function is related to its second derivative, as the curvature at a point increases when the absolute value of the second derivative increases.
The second derivative of y = 4ex is y'' = 4ex, which is always positive and increases as x increases, meaning that the curvature increases with x. Therefore, there is no maximum curvature in the domain of the function, as it will keep increasing as x goes towards infinity. However, for the given options A) x = 0, B) x = 1, C) x = -1, and D) x = 2, the curvature is greatest at the highest value of x, hence at x = 2.