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\n$f(x)=-2x^{3}+27x^{2}-108x$ 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 your choice\na. the absolute maximum/maxima is/are - 108 at $x = 6$\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\nwhat is/are the absolute minimum/minima of f on the given interval? select the correct choice below and, if necessary, fill in the answer boxes to complete your choice\na. the absolute minimum/minima 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 minimum of f on the given interval

determine the location and value of the absolute extreme values of f on the given interval, if they exist\n$f(x)=-2x^{3}+27x^{2}-108x$ 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 your choice\na. the absolute maximum/maxima is/are - 108 at $x = 6$\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\nwhat is/are the absolute minimum/minima of f on the given interval? select the correct choice below and, if necessary, fill in the answer boxes to complete your choice\na. the absolute minimum/minima 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 minimum of f on the given interval

Answer

Explanation:

Step1: Find the derivative of (f(x))

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). The critical points are (x = 3) and (x=6). Both (x = 3) and (x = 6) are in the interval ([2,7])

Step3: Evaluate (f(x)) at the critical points and endpoints

  • 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:

For the absolute maximum: A. The absolute maximum/maxima is/are (-108) at (x = 6) For the absolute minimum: A. The absolute minimum/minima is/are (-135) at (x = 3)