Determine the location and values of the absolute maximum and absolute minimum of given function: f(x)= ( -x+2)^4, Where 0<=x<=3

The absolute maximum and the absolute minimum are (0, 16) and (2, 0).

Step-by-step explanation:

First, we obtain the first and second derivatives of the function by chain rule and derivative for a power function, that is:

First derivative

Second derivative

Then, we proceed to do the First and Second Derivative Tests:

First Derivative Test

Second Derivative Test

The Second Derivative Test is unable to determine the nature of the critical values.

Then, we plot the function with the help of a graphing tool. The absolute maximum and the absolute minimum are (0, 16) and (2, 0).