consider the function with the following properties and answer the question below

Question

consider the function f(x)=-(x²-4)/(x²+1) with the following properties:

1. f has x-intercepts at x=±2 and a y-intercept at y=4
2. f has a horizontal asymptote at y=-1 and no vertical asymptotes
3. f, f', and f" are defined for all values of x
4. f'(x) is positive on (-∞,0) and negative on (0,∞)
5. f"(x) is positive on the open intervals (-∞,-1/√3) and (1/√3, ∞).
6. f"(x) is negative on the open interval (-1/√3,1/√3)
7. Does f have any local extrema? If so, state what type they are and the x-value where they occur.
8. Does the graph of f have any inflection points? If so, give the coordinates

Answer 1

My first suggestion is to of course: graph the function! This makes things easier to understand, especially if you are a visual learner. If this question is given on an exam, where you are not allowed a graphing calculator, then this tutorial should still be helpful for the sans-visual version of answering these questions. The following image was made using Desmos (https://www.desmos.com/calculator):

https://drive.google.com/file/d/1Y3LDy_C4dfVM9PsLk8XwNVha2wTWaqTS/view?usp=sharing

1) We see the function overall is essentially increasing for x<0 and decreasing x>0, this was also noted in bullet point 4. The function approaches that horizontal asymptote of y= -1 in the negative and positive infinite limits. Thus, there is only one local extremum, and that is the maximum occurring at (x, y) = (0,4). (This is also the global maximum.)

2) How do we recognize inflection points on a graph? Observe the curvature. Is it "smiling" (positive f'' ) or "frowning" (negative f'' ), and does this change in different portions of the function?

It is easy to see that positive curvature on the outer portions of the graph while the portion about x=0 is curved downward about that global maximum (negative curvature). The necessary information is also given in bullet points 5 and 6 of the problem statement. The sign of f'' changes, and thus we do have inflection points. Specifically, they occur at:

(x, y) = (-1/√3,2.75) and (x, y) = (1/ √3,2.75)

I hope this is helpful, let me know if you have questions!