Derivative and concavity

Theorem (derivative and concavity).
Let f be a function continuous on an interval I and twice differentiable on its interior Int(I ).
If f ′′ ≥ 0 on Int(I ), then f is concave up on I.
If f ′′ ≤ 0 on Int(I ), then f is concave down on I.

If f is concave up on I, then f ′′ ≥ 0 on Int(I ).
If f is concave down on I, then f ′′ ≤ 0 on Int(I ).

Just like in the previous section on monotonicity, we had to restrict ourself to connected sets, that is, to intervals.

For identifying points of inflection we can use the classification theorem from the previous section:

Theorem.
Let a be a point from the domain of a function f, assume that f has all derivatives there and f ′(a) = 0. Let n be the smallest integer for which f ⁽ⁿ⁾(a) is not zero.
If n is odd, then f has a point of inflection at a.

However, in practice it is easier to get this info from intervals of concavity. For this and other practical information see Concavity in Theory - Graphing functions.

Tricks with MVT
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