Context: 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 [-5,5] Answer question in terms of (a)^(4/3)

To find the absolute extreme values of a function on a given interval, we need to find the critical points of the function within the interval, evaluate the function at these points and at the endpoints of the interval, and compare these values.

Step 1: Find the derivative of the function. The derivative of f(x) = (x-2)^(4/3) is f'(x) = (4/3)(x-2)^(1/3).

Step 2: Find the critical points. Set the derivative equal to zero and solve for x: (4/3)(x-2)^(1/3) = 0 This implies that x = 2 is a critical point.

Step 3: Evaluate the function at the critical points and at the endpoints of the interval. f(2) = (2-2)^(4/3) = 0 f(-5) = ((-5)-2)^(4/3) = (-7)^(4/3) f(5) = (5-2)^(4/3) = 3^(4/3)

Step 4: Compare these values. The smallest value is 0 at x = 2, and the largest value is 3^(4/3) at x = 5.

So, the absolute minimum value of f on the interval [-5,5] is 0 at x = 2, and the absolute maximum value is 3^(4/3) at x = 5.