determine the location and value of the absolute extreme values of f on the given interval, if they…

determine the location and value of the absolute extreme values of f on the given interval, if they exist.\nf(x)=-2x^{3}+27x^{2}-108x on 2,7\nwhat is/are the absolute maximum/maxima of f on the given interval? select the correct choice below and, if necessary, fill in the answer boxes to complete you\na. the absolute maximum/maxima is/are at x=\n(use a comma to separate answers as needed. type exact answers, using radicals as needed.)\nb. there is no absolute maximum of f on the given interval.

Answer

Explanation:

Step1: Find the derivative of the function

The derivative of (f(x)=-2x^{3}+27x^{2}-108x) is (f^{\prime}(x)=-6x^{2}+54x - 108). Factor (f^{\prime}(x)): (f^{\prime}(x)=-6(x^{2}-9x + 18)=-6(x - 3)(x - 6)).

Step2: Find the critical points

Set (f^{\prime}(x)=0), then (-6(x - 3)(x - 6)=0). Solving (x - 3=0) gives (x = 3), and solving (x - 6=0) gives (x = 6). Both (x = 3) and (x = 6) are in the interval ([2,7]).

Step3: Evaluate the function at the critical points and endpoints

Evaluate (f(x)) at (x = 2), (x = 3), (x = 6), and (x = 7).

When (x = 2): (f(2)=-2\times2^{3}+27\times2^{2}-108\times2=-16 + 108-216=-124).
When (x = 3): (f(3)=-2\times3^{3}+27\times3^{2}-108\times3=-54 + 243-324=-135).
When (x = 6): (f(6)=-2\times6^{3}+27\times6^{2}-108\times6=-432+972 - 648=-108).
When (x = 7): (f(7)=-2\times7^{3}+27\times7^{2}-108\times7=-686+1323-756=-119).

Answer:

A. The absolute maximum/maxima is/are (-108) at (x = 6).