`f(x) = x^3 - 6x^2 + 9x + 1, [2,4]` Find the local and absolute extreme values of the function on the given interval.
Now to find the absolute extreme values of the function , that is continuous on a closed interval, we have to find the critical numbers that are in the interval and evaluate the function at the endpoints and at the critical numbers.
Now to find the critical numbers, solve for x for f'(x)=0.
However x=1 is not in the domain of the function, so the critical point is at x=3,
Now let's evaluate the function at the end points and at the critical number,
Absolute Maximum=5 at x=4
Absolute Minimum=1 at x=3