graph the function y = 2x^4 + 4x^3 = 2x^3(x + 2) by identifying the domain and any symmetries, finding the…

graph 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. identify where the curve is concave up or concave down. select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. a. the curve is never concave up and is concave down on the interval(s) (type your answer in interval notation. round to the nearest tenth as needed. use a comma to separate answers as needed.) b. the curve is concave up on the interval(s) and is concave down on the interval(s) (type your answer in interval notation. round to the nearest tenth as needed. use a comma to separate answers as needed.) c. the curve is concave up on the interval(s) and is never concave down (type your answer in interval notation. round to the nearest tenth as needed. use a comma to separate answers as needed.) d. the curve is neither concave up nor concave down. find any vertical asymptotes. select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.

graph 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. identify where the curve is concave up or concave down. select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. a. the curve is never concave up and is concave down on the interval(s) (type your answer in interval notation. round to the nearest tenth as needed. use a comma to separate answers as needed.) b. the curve is concave up on the interval(s) and is concave down on the interval(s) (type your answer in interval notation. round to the nearest tenth as needed. use a comma to separate answers as needed.) c. the curve is concave up on the interval(s) and is never concave down (type your answer in interval notation. round to the nearest tenth as needed. use a comma to separate answers as needed.) d. the curve is neither concave up nor concave down. find any vertical asymptotes. select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.

Answer

Explanation:

Step1: Find the first - derivative

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

Step2: Find the second - derivative

Differentiate $y'$ with respect to $x$. $y''=24x^{2}+24x=24x(x + 1)$.

Step3: Determine concavity

Set $y'' = 0$, then $24x(x + 1)=0$, which gives $x=0$ and $x=-1$. Test the intervals:

  • For $x<-1$, let $x=-2$. Then $y''=24\times(-2)\times(-2 + 1)=48>0$, so the curve is concave up on $(-\infty,-1)$.
  • For $-1<x<0$, let $x =-\frac{1}{2}$. Then $y''=24\times(-\frac{1}{2})\times(-\frac{1}{2}+1)=-6<0$, so the curve is concave down on $(-1,0)$.
  • For $x>0$, let $x = 1$. Then $y''=24\times1\times(1 + 1)=48>0$, so the curve is concave up on $(0,\infty)$.

Step4: Check for vertical asymptotes

Since $y = 2x^{4}+4x^{3}$ is a polynomial function (degree 4), the domain is $(-\infty,\infty)$ and there are no vertical asymptotes.

Answer:

For concavity: B. The curve is concave up on the interval(s) $(-\infty,-1),(0,\infty)$ and is concave down on the interval(s) $(-1,0)$. For vertical asymptotes: There are no vertical asymptotes.