Concavity

There are various terms used when the values of a function, its average rates of change, or its second average rates of change (the rates of change of the rates of change) are all positive (or negative), at least on some interval. When the function is differentiable, and especially when it's twice differentiable, there are easier ways to describe these. This is all summarized in the table below.
Property of fDefinition If differentiableIf twice differentiable
Positivef(a) > 0
Negativef(a) < 0
Increasing (f(b) − f(a)) ÷ (b − a) > 0 f′(a) > 0
Decreasing (f(b) − f(a)) ÷ (b − a) < 0 f′(a) < 0
Concave upward ((f(c) − f(b)) ÷ (c − b) − (f(b) − f(a)) ÷ (b − a)) ÷ (c − a) > 0 (f′(b) − f′(a)) ÷ (b − a) > 0 f″(a) > 0
Concave downward ((f(c) − f(b)) ÷ (c − b) − (f(b) − f(a)) ÷ (b − a)) ÷ (c − a) < 0 (f′(b) − f′(a)) ÷ (b − a) < 0 f″(a) < 0
In all of these, the function f has the given property on some interval if the given condition holds whenever a, b, and c are distinct numbers in that interval. (They must be distinct to avoid division by zero.)

Generally, it's much easier to work with the rightmost condition for every property, but you can't do that if the necessary derivatives don't exist. Even if the function isn't differentiable at all, it still makes sense to say whether it's concave upward or downward (or not).

Incidentally, here is some other terminology that you may see for these properties:


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