Context: Determine the location and value of the absolute extreme values of f on the given​ interval, if they exist. f(x) = (x-2)^(4/3) on [-7,7]

To find the absolute extreme values of a function on a given interval, we need to follow these steps:

1. Find the derivative of the function.
2. Set the derivative equal to zero and solve for x to find critical points.
3. Evaluate the function at the critical points and at the endpoints of the interval.
4. The largest and smallest values are the absolute maximum and minimum, respectively.

Let's apply these steps to the function f(x) = (x-2)^(4/3) on the interval [-7,7].

1. The derivative of f(x) = (x-2)^(4/3) is f'(x) = (4/3)(x-2)^(1/3).
2. Setting f'(x) = 0 gives (4/3)(x-2)^(1/3) = 0. Solving for x gives x = 2. So, 2 is a critical point.
3. Now, evaluate the function at the critical point and at the endpoints of the interval:
   • f(-7) = ((-7)-2)^(4/3) = (-9)^(4/3) = 81
   • f(2) = (2-2)^(4/3) = 0
   • f(7) = (7-2)^(4/3) = 5^(4/3) = 125^(1/3) = 5
4. Comparing these values, we see that the absolute maximum is 81 at x = -7, and the absolute minimum is 0 at x = 2.