Final answer:
To find the absolute extreme values of f on the given interval, we need to find the critical points and check the endpoints. The 10th derivative of f at x = 0 can be found by repeatedly differentiating f'(x) and evaluating the result at x = 0.
Step-by-step explanation:
To determine the location and value of the absolute extreme values of f on the interval [0.2, 3], we first need to find the critical points of the function f(x) = cos(4x²) - 1/x². The critical points occur where the derivative of the function is equal to zero or does not exist. We can find the derivative of f(x) using the chain rule: f'(x) = -8xcos(4x²) + 2/x³. To find the critical points, we set f'(x) equal to zero and solve for x.
Next, we check the endpoints of the interval [0.2, 3] to see if they yield the maximum or minimum values of f(x). To do this, we substitute the values of x = 0.2 and x = 3 into the function f(x) and compare the results with the values obtained from the critical points. The highest and lowest values obtained from the critical points and endpoints are the absolute maximum and minimum values of f(x) on the interval.
As for finding the 10th derivative of f at x = 0, we can use the power rule and chain rule repeatedly to find higher-order derivatives. The 10th derivative of f(x) can be obtained by finding the derivative of f'(x) 9 times and then evaluating the result at x = 0.