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
[tex]f'(x) = -4\cdot (-x+2)^{3}[/tex]
Second derivative
[tex]f''(x) = 12\cdot (-x + 2)^{2}[/tex]
Then, we proceed to do the First and Second Derivative Tests:
First Derivative Test
[tex]-4\cdot (-x+2)^{3} = 0[/tex]
[tex]-x + 2 = 0[/tex]
[tex]x = 2[/tex]
Second Derivative Test
[tex]f''(2) = 12\cdot (-2+2)^{2}[/tex]
[tex]f''(2) = 0[/tex]
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).
