Final answer:
The absolute maximum value of f(x) = x^3 - 12x^2 + 21x + 1 on the interval [0,9] is 343, which occurs at x = 7.
Step-by-step explanation:
The absolute maximum value of a function can be found by determining the critical points and evaluating the function at these points as well as the endpoints of the interval.
To find the critical points, we can take the derivative of the function and set it equal to zero:
f'(x) = 3x^2 - 24x + 21
Setting the derivative equal to zero and solving for x, we find x = 1 and x = 7. Plugging these values back into the original function, we find that f(1) = 11 and f(7) = 343.
Now we need to evaluate the function at the endpoints of the interval. f(0) = 1 and f(9) = 10.
Comparing all these values, we find that the absolute maximum value of f(x) = x^3 - 12x^2 + 21x + 1 on the interval [0,9] is 343, which occurs at x = 7.