Question

answer the following questions about the function whose derivative is ( f^{prime}(x)=2x(x + 4) ).
a. what are the critical points of ( f )?
b. on what open intervals is ( f ) increasing or decreasing?
c. at what points, if any, does ( f ) assume local maximum and minimum values?
a. find the critical points, if any. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the critical point(s) of ( f ) is/are ( x = ).
(simplify your answer. use a comma to separate answers as needed.)
b. the function ( f ) has no critical points.

Explanation:

Step1: Recall the definition of critical points

Critical points of a function \(y = f(x)\) are the values of \(x\) where \(f^{\prime}(x)=0\) or \(f^{\prime}(x)\) is undefined.
Given \(f^{\prime}(x)=2x(x + 4)\), and since \(f^{\prime}(x)\) is a polynomial (and polynomials are defined for all real \(x\), i.e., \(x\in(-\infty,\infty)\)), we only need to solve \(f^{\prime}(x)=0\).

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

Set \(2x(x + 4)=0\).
Using the zero - product property \(ab = 0\) implies \(a = 0\) or \(b=0\).
If \(2x=0\), then \(x = 0\); if \(x + 4=0\), then \(x=-4\).

Answer:

A. The critical point(s) of \(f\) is/are \(x=-4,0\)