Final answer:
To find the largest value of x³(1-x) on the interval 0 to 1, take the derivative and solve for critical points. Evaluate the function at critical points and endpoints to find the largest value.
Step-by-step explanation:
To find the largest value of x³(1-x) on the interval 0 to 1, we can first find the critical points of the function by taking the derivative and setting it equal to zero. The derivative of x³(1-x) is 3x² - 4x² + 1. Setting it equal to zero gives us the equation 3x² - 4x + 1 = 0. We can solve this equation to find the critical points. By factoring or using the quadratic formula, we find that x = 1/3 and x = 1 are the critical points. Next, we evaluate the function at these critical points as well as at the endpoints of the interval (x = 0 and x = 1). We find that the largest value of x³(1-x) on the interval 0 to 1 is 1/12, which occurs at x = 1/3.