Question

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

Answer 1

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

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

`f'(x) = -4\cdot (-x+2)^(3)`

Second derivative

`f''(x) = 12\cdot (-x + 2)^(2)`

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

First Derivative Test

`-4\cdot (-x+2)^(3) = 0`

`-x + 2 = 0`

`x = 2`

Second Derivative Test

`f''(2) = 12\cdot (-2+2)^(2)`

`f''(2) = 0`

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).