Question

Let f(x)=-6 ln (x+2)/x+2+2 Determine the absolute extrema of f on [-1,3]. If multiple such values exist, enter the solutions using a comma-separated list.

Answer 1

To find the absolute extrema of the function f(x) = -6 ln(x + 2) / (x + 2) + 2 on the interval [-1,3], determine its critical points, evaluate the function at these points, and also at the endpoints of the interval to find the largest and smallest values, representing the absolute maximum and minimum.

Step-by-step explanation:

To determine the absolute extrema of the function f(x) = -6 ln(x + 2) / (x + 2) + 2 on the interval [-1,3], we first have to find the critical points within this interval and then evaluate the function at these critical points and at the endpoints of the interval.

To find the critical points, we find the derivative of f(x), set it equal to zero and solve for x. It's important to note that the critical points must be within the interval [-1,3]. After finding the critical points, we must evaluate f(x) at these points as well as at the endpoints x = -1 and x = 3. The largest and smallest values obtained from these evaluations will be the absolute maximum and minimum, respectively, on the given interval.