Question

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. find any oblique asymptotes. select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. a. the function has one oblique asymptote, (type an equation.) b. the function has two oblique asymptotes. the asymptote with smaller slope is and the asymptote with larger slope is (type equations.) c. the function has no oblique asymptotes.

Explanation:

Step1: Check degree of numerator and denominator

For a rational function \(y = \frac{f(x)}{g(x)}\) where \(f(x)=2x\) (degree \(n = 1\)) and \(g(x)=x^{2}-16\) (degree \(m=2\)).

Step2: Apply oblique asymptote rule

A rational function \(y=\frac{f(x)}{g(x)}\) has an oblique asymptote if \(n=m - 1\). Here \(n=1\) and \(m = 2\), but for oblique asymptote we perform polynomial long - division. However, since \(n

Answer:

C. The function has no oblique asymptotes.