summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of y =…

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of y = f(x). f(x)=2x(x - 3)^3 what is/are the local maximum/a? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the local maximum/a is/are at x = (type an integer or simplified fraction. use a comma to separate answers as needed.) b. there is no local maximum. what is/are the local minimum/a? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the local minimum/a is/are at x = 3/4 (type an integer or simplified fraction. use a comma to separate answers as needed.) b. there is no local minimum. what are the inflection points? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the inflection points are at x = (type an integer or simplified fraction. use a comma to separate answers as needed.) b. there are no inflection points.

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of y = f(x). f(x)=2x(x - 3)^3 what is/are the local maximum/a? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the local maximum/a is/are at x = (type an integer or simplified fraction. use a comma to separate answers as needed.) b. there is no local maximum. what is/are the local minimum/a? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the local minimum/a is/are at x = 3/4 (type an integer or simplified fraction. use a comma to separate answers as needed.) b. there is no local minimum. what are the inflection points? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the inflection points are at x = (type an integer or simplified fraction. use a comma to separate answers as needed.) b. there are no inflection points.

Answer

Explanation:

Step1: Find the first - derivative

Use the product rule $(uv)^\prime = u^\prime v+uv^\prime$, where $u = 2x$ and $v=(x - 3)^3$. $u^\prime=2$ and $v^\prime = 3(x - 3)^2$. $f^\prime(x)=2(x - 3)^3+2x\times3(x - 3)^2=2(x - 3)^2[(x - 3)+3x]=2(x - 3)^2(4x - 3)$. Set $f^\prime(x)=0$, then $(x - 3)^2(4x - 3)=0$. The critical points are $x = 3$ and $x=\frac{3}{4}$. Use the first - derivative test. Consider the intervals $(-\infty,\frac{3}{4})$, $(\frac{3}{4},3)$ and $(3,\infty)$. For $x\in(-\infty,\frac{3}{4})$, let $x = 0$, $f^\prime(0)=2\times(- 3)^2\times(-3)<0$. For $x\in(\frac{3}{4},3)$, let $x = 1$, $f^\prime(1)=2\times(-2)^2\times1>0$. For $x\in(3,\infty)$, let $x = 4$, $f^\prime(4)=2\times1^2\times13>0$. So $x=\frac{3}{4}$ is a local minimum and there is no local maximum.

Step2: Find the second - derivative

Use the product rule on $f^\prime(x)=2(x - 3)^2(4x - 3)$. Let $u = 2(x - 3)^2$ and $v = 4x - 3$. $u^\prime=4(x - 3)$ and $v^\prime = 4$. $f^{\prime\prime}(x)=4(x - 3)(4x - 3)+2(x - 3)^2\times4=4(x - 3)[(4x - 3)+2(x - 3)]=4(x - 3)(6x - 9)=12(x - 3)(2x - 3)$. Set $f^{\prime\prime}(x)=0$, then $(x - 3)(2x - 3)=0$. Solving gives $x = 3$ and $x=\frac{3}{2}$.

Answer:

What is/are the local maximum/a? B. There is no local maximum. What is/are the local minimum/a? A. The local minimum/a is/are at $x=\frac{3}{4}$. What are the inflection points? A. The inflection points are at $x=\frac{3}{2},3$.