this question: 2 point(s) possible\ngraph the function y = 2x^4 + 4x^3 = 2x^3(x + 2) by identifying the…

this question: 2 point(s) possible\ngraph the function y = 2x^4 + 4x^3 = 2x^3(x + 2) by identifying the domain and any symmetries, finding the derivatives y and y, finding the critical points and identifying the functions behavior at each one, finding where the curve is increasing and where it is decreasing, finding the points of inflection, determining the concavity of the curve, identifying any asymptotes, and plotting any key points such as intercepts, critical points, and inflection points. then find coordinates of absolute extreme points, if any.\nidentify where the curve is increasing or decreasing. select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.\na. the curve increases on the open interval(s) and does not decrease.\n(type your answer in interval notation. round to the nearest tenth as needed. use a comma to separate answers as needed.)\nb. the curve does not increase and decreases on the open interval(s).\n(type your answer in interval notation. round to the nearest tenth as needed. use a comma to separate answers as needed.)\nc. the curve increases on the open interval(s) and decreases on the open interval(s).\n(type your answer in interval notation. round to the nearest tenth as needed. use a comma to separate answers as needed.)\nd. the curve neither increases nor decreases.\nidentify any inflection points. select the correct choice below and, if necessary, fill in the answer box to complete your choice.

this question: 2 point(s) possible\ngraph the function y = 2x^4 + 4x^3 = 2x^3(x + 2) by identifying the domain and any symmetries, finding the derivatives y and y, finding the critical points and identifying the functions behavior at each one, finding where the curve is increasing and where it is decreasing, finding the points of inflection, determining the concavity of the curve, identifying any asymptotes, and plotting any key points such as intercepts, critical points, and inflection points. then find coordinates of absolute extreme points, if any.\nidentify where the curve is increasing or decreasing. select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.\na. the curve increases on the open interval(s) and does not decrease.\n(type your answer in interval notation. round to the nearest tenth as needed. use a comma to separate answers as needed.)\nb. the curve does not increase and decreases on the open interval(s).\n(type your answer in interval notation. round to the nearest tenth as needed. use a comma to separate answers as needed.)\nc. the curve increases on the open interval(s) and decreases on the open interval(s).\n(type your answer in interval notation. round to the nearest tenth as needed. use a comma to separate answers as needed.)\nd. the curve neither increases nor decreases.\nidentify any inflection points. select the correct choice below and, if necessary, fill in the answer box to complete your choice.

Answer

Explanation:

Step1: Find the first - derivative

Given $y = 2x^{4}+4x^{3}=2x^{3}(x + 2)=2x^{4}+4x^{3}$. Using the power rule $(x^n)'=nx^{n - 1}$, we have $y'=8x^{3}+12x^{2}=4x^{2}(2x + 3)$.

Step2: Find the critical points

Set $y'=0$. Then $4x^{2}(2x + 3)=0$. Solving $4x^{2}=0$ gives $x = 0$, and solving $2x+3=0$ gives $x=-\frac{3}{2}$.

Step3: Determine the intervals of increase and decrease

We consider the intervals $(-\infty,-\frac{3}{2})$, $(-\frac{3}{2},0)$ and $(0,\infty)$. For $x\in(-\infty,-\frac{3}{2})$, let's take a test - point $x=-2$. Then $y'=4(-2)^{2}[2(-2)+3]=4\times4\times(-1)=-16<0$, so the function is decreasing on $(-\infty,-\frac{3}{2})$. For $x\in(-\frac{3}{2},0)$ and $x\in(0,\infty)$, let's take test - points $x =-\frac{1}{2}$ and $x = 1$ respectively. When $x=-\frac{1}{2}$, $y'=4(-\frac{1}{2})^{2}[2(-\frac{1}{2})+3]=4\times\frac{1}{4}\times2 = 2>0$. When $x = 1$, $y'=4\times1^{2}(2\times1+3)=4\times5 = 20>0$. So the function is increasing on $(-\frac{3}{2},\infty)$.

Step4: Find the second - derivative

$y'=8x^{3}+12x^{2}$, then $y'' = 24x^{2}+24x=24x(x + 1)$.

Step5: Find the inflection points

Set $y''=0$. Then $24x(x + 1)=0$. Solving gives $x = 0$ and $x=-1$.

Answer:

For the increasing - decreasing part: C. The curve increases on the open interval(s) $(-\frac{3}{2},\infty)$ and decreases on the open interval(s) $(-\infty,-\frac{3}{2})$. For the inflection points: The inflection points are at $x=-1$ and $x = 0$. To find the $y$ - coordinates, when $x=-1$, $y=2(-1)^{4}+4(-1)^{3}=2 - 4=-2$; when $x = 0$, $y=0$. So the inflection points are $(-1,-2)$ and $(0,0)$.