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) $5,8$
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 $-24$ at $x = 4$
(use a comma to separate answers as needed.)
b. there is no absolute minimum.
(b) 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 at $x=$
(use a comma to separate answers as needed.)
b. there is no absolute maximum.

Explanation:

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

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

Step2: Find critical points

Set \(f^\prime(x)=0\).
Since \(4(x - 7)^2(x - 4)=0\), then \(x = 4\) or \(x = 7\)

Step3: Evaluate \(f(x)\) for interval \([2,9]\)

• At \(x = 2\): \(f(2)=(2 - 3)(2 - 7)^3+3=(-1)\times(-125)+3 = 128\)
• At \(x = 4\): \(f(4)=(4 - 3)(4 - 7)^3+3=1\times(-27)+3=-24\)
• At \(x = 7\): \(f(7)=(7 - 3)(7 - 7)^3+3=3\)
• At \(x = 9\): \(f(9)=(9 - 3)(9 - 7)^3+3=6\times8 + 3=51\)

Answer:

The absolute maximum is \(128\) at \(x = 2\)