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. graph the function y = 2x^4 + 4x^3. choose the correct graph.

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. graph the function y = 2x^4 + 4x^3. choose the correct graph.

Answer

Answer:

To graph the function (y = 2x^{4}+4x^{3}), we follow these steps:

Step1: Find the domain

The domain of a polynomial function is all real - numbers. So, the domain of (y = 2x^{4}+4x^{3}) is ((-\infty,\infty)).

Step2: Check for symmetries

  • Even function: (y(-x)=2(-x)^{4}+4(-x)^{3}=2x^{4}-4x^{3}\neq y(x)), so it is not an even function.
  • Odd function: (y(-x)=2x^{4}-4x^{3}\neq - y(x)=-2x^{4}-4x^{3}), so it is not an odd function. There is no symmetry about the (y) - axis or the origin.

Step3: Find the first - derivative

Using the power rule ((x^{n})^\prime=nx^{n - 1}), we have (y^\prime=\frac{d}{dx}(2x^{4}+4x^{3})=8x^{3}+12x^{2}=4x^{2}(2x + 3)).

Step4: Find the critical points

Set (y^\prime = 0), then (4x^{2}(2x + 3)=0).

  • (4x^{2}=0) gives (x = 0).
  • (2x+3=0) gives (x=-\frac{3}{2}).

Step5: Determine intervals of increase and decrease

  • Choose test points: Let's choose (x=-2), (x =-\frac{1}{2}), and (x = 1).
    • When (x=-2), (y^\prime=4(-2)^{2}(2(-2)+3)=16(-1)=-16<0), so the function is decreasing on the interval ((-\infty,-\frac{3}{2})).
    • When (x =-\frac{1}{2}), (y^\prime=4(-\frac{1}{2})^{2}(2(-\frac{1}{2})+3)=4\times\frac{1}{4}(2)=2>0), so the function is increasing on the interval ((-\frac{3}{2},0)\cup(0,\infty)).

Step6: Find the second - derivative

(y^{\prime\prime}=\frac{d}{dx}(8x^{3}+12x^{2})=24x^{2}+24x = 24x(x + 1)).

Step7: Find the inflection points

Set (y^{\prime\prime}=0), then (24x(x + 1)=0).

  • (x = 0) and (x=-1) are the inflection points.

Step8: Determine concavity

  • Choose test points: Let (x=-2), (x=-\frac{1}{2}), and (x = 1).
    • When (x=-2), (y^{\prime\prime}=24(-2)(-2 + 1)=48>0), so the function is concave up on ((-\infty,-1)).
    • When (x=-\frac{1}{2}), (y^{\prime\prime}=24(-\frac{1}{2})(-\frac{1}{2}+1)=-6<0), so the function is concave down on ((-1,0)).
    • When (x = 1), (y^{\prime\prime}=24\times1\times(1 + 1)=48>0), so the function is concave up on ((0,\infty)).

Step9: Find the (y) - intercept

Set (x = 0), then (y=2(0)^{4}+4(0)^{3}=0).

Step10: Find the (x) - intercepts

Set (y = 0), then (2x^{4}+4x^{3}=2x^{3}(x + 2)=0). So (x = 0) and (x=-2) are the (x) - intercepts.

Since it is a polynomial function of degree 4 with a positive leading coefficient ((a = 2>0)), as (x\to\pm\infty), (y\to\infty).

Based on the above - mentioned analysis of the function's properties (intercepts, critical points, intervals of increase/decrease, concavity, and end - behavior), we can plot the graph. Without seeing the actual options clearly, we have analyzed the function (y = 2x^{4}+4x^{3}) step - by - step.

There is no final answer value given as the question only asks for the steps to graph the function and does not provide enough information about the multiple - choice options to select one. But the above steps are sufficient to graph the function.