answer the following questions about the function whose derivative is f(x)=3x(x + 4). a. what are the…

answer the following questions about the function whose derivative is f(x)=3x(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. o a. the critical point(s) of f is/are x = (simplify your answer. use a comma to separate answers as needed.) o b. the function f has no critical points. b. determine where f is increasing and decreasing. select the correct choice below and fill in the answer box to complete your choice. (type your answer in interval notation. use a comma to separate answers as needed.) o a. the function f is decreasing on the open interval(s) , and never increasing. o b. the function f is increasing on the open interval(s) , and decreasing on the open interval(s) o c. the function f is increasing on the open interval(s) , and never decreasing. c. determine the local maximum/maxima, if any. select the correct choice below and, if necessary, fill in the answer box to complete your choice.

answer the following questions about the function whose derivative is f(x)=3x(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. o a. the critical point(s) of f is/are x = (simplify your answer. use a comma to separate answers as needed.) o b. the function f has no critical points. b. determine where f is increasing and decreasing. select the correct choice below and fill in the answer box to complete your choice. (type your answer in interval notation. use a comma to separate answers as needed.) o a. the function f is decreasing on the open interval(s) , and never increasing. o b. the function f is increasing on the open interval(s) , and decreasing on the open interval(s) o c. the function f is increasing on the open interval(s) , and never decreasing. c. determine the local maximum/maxima, if any. select the correct choice below and, if necessary, fill in the answer box to complete your choice.

Answer

Explanation:

Step1: Find critical points

Set $f'(x)=0$, so $3x(x + 4)=0$. Then $x=0$ or $x=-4$.

Step2: Determine increasing - decreasing intervals

Create intervals $(-\infty,-4)$, $(-4,0)$ and $(0,\infty)$. Test a value in each interval in $f'(x)$. For $x=-5$, $f'(-5)=3\times(-5)\times(-5 + 4)=15>0$. For $x=-1$, $f'(-1)=3\times(-1)\times(-1 + 4)=-9<0$. For $x = 1$, $f'(1)=3\times1\times(1 + 4)=15>0$. So $f$ is increasing on $(-\infty,-4)\cup(0,\infty)$ and decreasing on $(-4,0)$.

Step3: Find local maxima and minima

Since $f$ changes from increasing to decreasing at $x=-4$, $f(-4)$ is a local maximum. Since $f$ changes from decreasing to increasing at $x = 0$, $f(0)$ is a local minimum.

Answer:

a. A. The critical point(s) of $f$ is/are $x=-4,0$ b. B. The function $f$ is increasing on the open interval(s) $(-\infty,-4),(0,\infty)$ and decreasing on the open interval(s) $(-4,0)$ c. The function has a local maximum at $x=-4$ and a local minimum at $x = 0$