Question

the first derivative test
question
part 2 of 4
completed: 0 of 3 my score: 0/3 pts (0%)
for the following questions about the function whose derivative is ( f^{prime}(x)=2 x(x + 4) ).
what are the critical points of ( f )?
what open intervals is ( f ) increasing or decreasing?
what points, if any, does ( f ) assume local maximum and minimum values?
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=-4,0 ).
(simplify your answer. use a comma to separate answers as needed.)
b. the function ( f ) has no critical points.
determine where ( f ) is increasing and decreasing. select the correct choice below and fill in the answer box to complete your choice.
pe your answer in interval notation. use a comma to separate answers as needed.)
a. the function ( f ) is increasing on the open interval(s), and never decreasing.
b. the function ( f ) is increasing on the open interval(s), and decreasing on the open interval(s).
c. the function ( f ) is decreasing on the open interval(s), and never increasing.

Explanation:

Step1: Find critical points

Critical points occur where \(f^{\prime}(x) = 0\). Given \(f^{\prime}(x)=2x(x + 4)\), set \(2x(x + 4)=0\).
Using the zero - product property \(ab = 0\) implies \(a = 0\) or \(b=0\). So \(2x=0\) gives \(x = 0\), and \(x + 4=0\) gives \(x=-4\).

Step2: Determine intervals of increase and decrease

We use test points in the intervals \((-\infty,-4)\), \((-4,0)\), and \((0,\infty)\).

• For the interval \((-\infty,-4)\), let \(x=-5\). Then \(f^{\prime}(-5)=2\times(-5)\times(-5 + 4)=10>0\).
• For the interval \((-4,0)\), let \(x=-1\). Then \(f^{\prime}(-1)=2\times(-1)\times(-1 + 4)=-6<0\).
• For the interval \((0,\infty)\), let \(x = 1\). Then \(f^{\prime}(1)=2\times1\times(1 + 4)=10>0\).

Since \(f^{\prime}(x)>0\) on \((-\infty,-4)\cup(0,\infty)\) and \(f^{\prime}(x)<0\) on \((-4,0)\)

Answer:

• Critical points: \(x=-4,0\)
• The function \(f\) is increasing on the open intervals \((-\infty,-4)\) and \((0,\infty)\), and decreasing on the open interval \((-4,0)\). So the answer for the second part is B with increasing intervals \((-\infty,-4),(0,\infty)\) and decreasing interval \((-4,0)\)