Question

What is concavity? How does it relate to the second derivative?

Answer 1

Some books define concavity as the sign of the second derivative over an interval. Namely "concave up" is defined as an interval where 2nd derivative is positive, concave down is defined as an interval where 2nd derivative is negative.

Other books define concavity separately from the sign of the second derivative

Answer 2

because first derivative is already taken as increasing or decreasing... XD

Answer 3

It seems like concavity is never defined. Most website says that concavity has the same sign with the second derivative but does not define concavity.

Answer 4

positive - increasing
negative decreasing
and then there's the infliction point in which the direction of the line changes direction... it's looks similar to symmetry on the original for an odd function XD

Answer 5

Here is my book definition
Suppose that the function f is differentiable at the point 'a' and that L is the line tangent to the graph y = f(x) at the point (a,f(a)). Then the function f is said to be
1. Concave upward at 'a' if , on some open interval containing a, the graph of f lies above L
2. Concave downward at 'a' if on some open interval containing 'a', the graph of f lies below L

Answer 6

Now there is a theorem. Suppose f is twice differentiable on an open interval I
1. if f ' ' (x) > 0 on I, then f is concave upward at each point of I.
2. if f ' ' (x) < 0 on I, then f is concave downward at each point of I.

Answer 7

or as I like to remember it as concave up happy face concave down sad face XD

Answer 8

but its easier to just define concave up as a spot where the the second deriv is positive, concave down is where second deriv is neg.
but technically concave upward means that the function lies above the tangent to the function at the point