Question

determine all critical points for the following function.
f(x)=2x(16 - x)^3
what is the derivative of f(x)=2x(16 - x)^3?
f(x)=8(16 - x)^2(4 - x)
find the critical points, if any, of f on the domain. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the critical point(s) occur(s) at x=
(use a comma to separate answers as needed.)
b. there are no critical points for f(x)=2x(16 - x)^3 on the domain.

Explanation:

Step1: Recall the definition of critical points

Critical points occur where \(f^{\prime}(x) = 0\) or \(f^{\prime}(x)\) is undefined. Since \(f^{\prime}(x)=8(16 - x)^{2}(4 - x)\) is a polynomial, it is defined for all real \(x\). So we only need to solve \(f^{\prime}(x)=0\).

Step2: Set \(f^{\prime}(x)=0\)

Set \(8(16 - x)^{2}(4 - x)=0\).
Using the zero - product property \(ab = 0\) implies \(a = 0\) or \(b = 0\).
If \(8(16 - x)^{2}(4 - x)=0\), then either \((16 - x)^{2}=0\) or \((4 - x)=0\).
For \((16 - x)^{2}=0\), we solve \(16 - x=0\), which gives \(x = 16\).
For \(4 - x=0\), we solve \(x = 4\).

Answer:

A. The critical point(s) occur(s) at \(x = 4,16\)