How do you find all points of inflection given #y=x^2/(2x+2)#?
Inflection points are the points where the graph's concavity changes—that is, where the slopes of the graph go from increasing to decreasing (or vice versa). Visually, inflection points are the points where the graph curves like a bowl on one side (concave up) and like a hill on the other (concave down). (e.g.
Mathematically, inflection points occur where the second derivative is equal to 0. So we need to find
(We can leave the numerator un-factored to help us find its derivative again.)
Now, we take this
We now set this second derivative equal to 0 and try to solve for
Uh-oh! We can't do it.
Because of this, there are no inflection points, but there will be a vertical asymptote when