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

The absolute maximum on the interval 0 < x < 3 is at x = 2 and f(2) = 0. Since x = 2 can only give an absolute maximum, so there is no absolute minimum.

Step-by-step explanation:

f(x) = (-x + 2)⁴

to find the absolute maximum and minimum values, we differentiate f(x) with respect to x.

So df(x)/dx = f'(x) = 4(-x + 2)³

The maximum and minimum values are obtained when f'(x) = 0

So,  4(-x + 2)³ = 0

⇒  (-x + 2)³ = 0

⇒ -x + 2 = 0

-x = -2

x = 2

Now, f(2) = (-2 + 2)⁴ = 0⁴ = 0

So, the absolute maximum on the interval 0 < x < 3 is at x = 2 and f(2) = 0. Since x = 2 can only give an absolute maximum, so there is no absolute minimum.