Final answer:
To find the maximum value of the function, we need to find the critical points, evaluate the function at these points and the endpoints, and compare the values. The maximum value of the function on the interval [0, 1] is 1, which contradicts the statement in the question that only about 30% of a river's energy can be captured.
Step-by-step explanation:
This question asks us to find the maximum value of the function E(x) = 2x³ - 4x² + 2x on the interval [0, 1]. To do this, we need to find the critical points of the function and evaluate the function at these points and the endpoints. The critical points are found by taking the derivative of the function with respect to x and setting it equal to zero. By solving for x, we find that there is one critical point at x = 0.5. We then substitute the critical point and the endpoints (x = 0 and x = 1) into the function and compare the values. By evaluating the function at x = 0, x = 0.5, and x = 1, we find that the maximum value of the function on the interval [0, 1] is 1. This means that the proportion of a river's energy that can be obtained from an undershot waterwheel is at most 1, or 100%. Therefore, this contradicts the statement in the question that only about 30% of a river's energy can be captured.