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

Answer:

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

$f'(x) = -4\cdot (-x+2)^{3}$

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).