Question

find the absolute maximum and minimum, if either exists, for the function on the indicated interval.
f(x)=(x - 3)(x - 7)^{3}+3
(a)0,5 (b)2,9 (c)5,8

(c) find the absolute maximum. select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
a. the absolute maximum is 8 at x = 8.
(use a comma to separate answers as needed.)
b. there is no absolute maximum.
find the absolute minimum. select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
a. the absolute minimum is at x=
(use a comma to separate answers as needed.)
b. there is no absolute minimum.

Explanation:

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

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

Step2: Find the critical points in the interval \([5,8]\)

Set \(f^\prime(x)=0\), \(4(x - 7)^2(x - 4)=0\) gives \(x = 4\) and \(x = 7\). But \(x = 4
otin[5,8]\), so the critical point in \([5,8]\) is \(x = 7\)

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

• For \(x = 5\): \(f(5)=(5 - 3)(5 - 7)^3+3=2\times(-8)+3=-16 + 3=-13\)
• For \(x = 7\): \(f(7)=(7 - 3)(7 - 7)^3+3=3\)
• For \(x = 8\): \(f(8)=(8 - 3)(8 - 7)^3+3=5\times1+3=8\)

Answer:

The absolute minimum is \(-13\) at \(x = 5\)