Question about concavity
The following three statements are basic facts about calculus.
f’ shows whether a function is increasing or decreasing.
f’’ shows whether a function is concave up or concave down.
the value of f’ shows the magnitude of its affects on f (like if f’ is very close to zero, f changes very slowly, but if f’ is very far away from zero, f changes very quickly)
This gives me two questions. I asked the first one to my calc BC teacher and she didn’t really have a good answer, and the second one is something I thought of more recently.
Since the value of f’ can be used to determine how much a function increases by at a point, and the signs of f’ and f’’ reflect similarly on the function f (either being increasing/concave up if positive or decreasing/concave down if negative), can f’‘ be used to infer “how concave” a function is?
This is probably a more typical question, but since the first and second derivatives of a function have visible effects on a graph, do any of the higher order derivatives have this same effect?