Question

Find the absolute extrema of the function on the indicated interval. Write your answers as ordered pairs with irrational numbers as needed (do not write approximations). g(x)=x-cos (x)

Answer 1

To find the absolute extrema of g(x) = x - cos(x) on the given interval, take the derivative, find the critical points, and evaluate the function at the endpoints. The absolute minimum is (0, 0) and the absolute maximum is (π, 2π - 1).

Step-by-step explanation:

To find the absolute extrema of the function g(x) = x - cos(x) on the given interval, we need to find the points where the function reaches its maximum and minimum values. This can be done by taking the derivative of g(x) and setting it equal to zero to find the critical points. We also need to evaluate the function at the endpoints of the interval. The highest point will be the absolute maximum and the lowest point will be the absolute minimum.

Taking the derivative of g(x) with respect to x, we get g'(x) = 1 + sin(x). Setting g'(x) = 0, we have sin(x) = -1. The solutions to this equation are x = (2n + 1)π, where n is an integer. Since the interval is 0 ≤ x ≤ 20, we only need to consider the value x = π as the critical point.

Next, we evaluate g(x) at the critical point x = π and the endpoints of the interval x = 0 and x = 20. The ordered pairs representing these points are (0, 0), (π, 2π - 1), and (20, 20 - cos(20)), respectively.

Comparing the y-values of these ordered pairs, we can determine that the absolute minimum is (0, 0) and the absolute maximum is (π, 2π - 1).