graph the function $y = \\frac{2x}{x^{2}-16}$ by identifying the domain and any symmetries, finding the…

graph the function $y = \\frac{2x}{x^{2}-16}$ 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. c. the curve increases on the open interval(s) and decreases on the open interval(s) (type your answer in interval notation. use a comma to separate answers as needed.) d. the curve neither increases nor decreases. identify any inflection points. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the inflection point(s) is/are at (type an ordered pair. use a comma to separate answers as needed.) b. there are no inflection points.

graph the function $y = \\frac{2x}{x^{2}-16}$ 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. c. the curve increases on the open interval(s) and decreases on the open interval(s) (type your answer in interval notation. use a comma to separate answers as needed.) d. the curve neither increases nor decreases. identify any inflection points. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the inflection point(s) is/are at (type an ordered pair. use a comma to separate answers as needed.) b. there are no inflection points.

Answer

Explanation:

Step1: Find the first derivative

Using the quotient rule ((\frac{u}{v})^\prime=\frac{u^\prime v - uv^\prime}{v^2}), where (u = 2x), (u^\prime=2), (v=x^{2}-16), (v^\prime = 2x). [ \begin{align*} y^\prime&=\frac{2(x^{2}-16)-2x\times(2x)}{(x^{2}-16)^{2}}\ &=\frac{2x^{2}-32 - 4x^{2}}{(x^{2}-16)^{2}}\ &=\frac{-2x^{2}-32}{(x^{2}-16)^{2}}\ &=\frac{-2(x^{2}+16)}{(x^{2}-16)^{2}} \end{align*} ] Since (x^{2}+16>0) and ((x^{2}-16)^{2}>0) for (x\neq\pm4), (y^\prime<0) for all (x) in the domain (\mathbb{R}\setminus{\pm4}).

Step2: Analyze increasing and decreasing intervals

The function is decreasing on ((-\infty,-4)), ((-4,4)) and ((4,\infty))

Step3: Find the second derivative

Using the quotient rule again, (u=-2(x^{2}+16)), (u^\prime=-4x), (v=(x^{2}-16)^{2}), (v^\prime = 4x(x^{2}-16)) [ \begin{align*} y^{\prime\prime}&=\frac{-4x(x^{2}-16)^{2}-(-2)(x^{2}+16)\times4x(x^{2}-16)}{(x^{2}-16)^{4}}\ &=\frac{-4x(x^{2}-16)+8x(x^{2}+16)}{(x^{2}-16)^{3}}\ &=\frac{-4x^{3}+64x + 8x^{3}+128x}{(x^{2}-16)^{3}}\ &=\frac{4x^{3}+192x}{(x^{2}-16)^{3}}\ &=\frac{4x(x^{2}+48)}{(x^{2}-16)^{3}} \end{align*} ] Set (y^{\prime\prime}=0), then (x = 0) (since (x^{2}+48>0) for all (x)). When (x<0,x\neq - 4), (y^{\prime\prime}<0) (concave down); when (x>0,x\neq4), (y^{\prime\prime}>0) (concave up)

Answer:

For the increasing - decreasing part: C. The curve increases on the open interval(s) and decreases on the open interval(s) ((-\infty,-4),(-4,4),(4,\infty)) For the inflection point: A. The inflection point(s) is/are at ((0,0))