G. CONCAVITY AND POINTS OF INFLECTION
Concavity and points of inflection were already discussed above. For the application of the information I provided above, analyze my solution on the left. To find the concavity of the given rational function, I applied the Second Derivative Test (SDT). I solved for the second derivative of the given function by differentiating the first derivative. In the process of differentiation, I followed the quotient rule since I am dealing with a rational function. Of course, one must remember all the other basic derivative rules such as the power rule, constant multiple rule, constant rule, and the sum and difference rule. After getting the second derivative, set it equal to zero and find its critical points. Note that critical points are the points where the second derivative is zero or does not exist. However, in this example, I used the same critical points I got from the first derivative test namely, 0, -2, and 2. Then, I got the intervals (-∞, -2), (-2, 0), (0, 2), and (2, ∞). From these intervals, I picked random number that is included in each interval such as -3, -1, 1, and 3, respectively. Then, I plugged in these values to the second derivative and observed the sign if positive or negative. If f" > 0 or positive, the interval is concave upward. If f" < 0 or negative, the interval is concave downward. In conclusion, I found out that the graph is concave upward at the intervals (-2, 0) and (0, 2) and is concave downward at the intervals (-∞, -2) and (2, ∞). Meanwhile, despite the change of concavity, I found out that the graph has no point of inflection. And it is because the original rational function is undefined to some values of x specifically 2 and -2. Note that if a function is undefined to a value of x, there will be a vertical asymptote that the graph approaches but never meets. Thus, there will be no point of inflection.