find the absolute maximum and minimum, if either exists, for the function on the indicated…

find the absolute maximum and minimum, if either exists, for the function on the indicated interval.\n\n$f(x)=(x - 3)(x - 7)^{3}+3$\n(a) $0,5$\n(b) $2,9$\n(c) $5,8$\n\n(a) find the absolute maximum. select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.\n\na. the absolute maximum is at $x=$\n(use a comma to separate answers as needed.)\n\nb. there is no absolute maximum.

find the absolute maximum and minimum, if either exists, for the function on the indicated interval.\n\n$f(x)=(x - 3)(x - 7)^{3}+3$\n(a) $0,5$\n(b) $2,9$\n(c) $5,8$\n\n(a) find the absolute maximum. select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.\n\na. the absolute maximum is at $x=$\n(use a comma to separate answers as needed.)\n\nb. there is no absolute maximum.

Answer

Explanation:

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

Use the product rule ((uv)^\prime = u^\prime v+uv^\prime), where (u=(x - 3)) and (v=(x - 7)^3). (u^\prime=1), (v^\prime = 3(x - 7)^2). So (t^\prime(x)=(x - 7)^3+3(x - 3)(x - 7)^2=(x - 7)^2[(x - 7)+3(x - 3)]=(x - 7)^2(4x-16)=4(x - 7)^2(x - 4))

Step2: Find the critical points

Set (t^\prime(x)=0), then (4(x - 7)^2(x - 4)=0). The critical points are (x = 4) and (x = 7). But for interval ([0,5]), (x = 7) is not in the interval.

Step3: Evaluate the function at critical points and endpoints

For (x = 0): (t(0)=(0 - 3)(0 - 7)^3+3=(-3)\times(-343)+3 = 1032) For (x = 4): (t(4)=(4 - 3)(4 - 7)^3+3=(1)\times(-27)+3=-24) For (x = 5): (t(5)=(5 - 3)(5 - 7)^3+3=(2)\times(-8)+3=-13)

Answer:

A. The absolute maximum is (1032) at (x = 0)