mathematics-physics-wiki/docs/en/mathematics/calculus/concavity-and-inflections.md

Concavity and inflections

Concave up

A function f is concave up on an open differentiable interval I if the derivative f' is an increasing function on I, then f'' > 0. Obtaining tangent line above the graph.

Concave dowm

A function f is concave down on an open and differentiable interval I if the derivative is a decreasing function on I, then f'' < 0. Obtaining tangent lines below the graph.

Inflection points

The function f has an inflection point at x_0 if

1. the tangent line in (x_0, f(x_0)) exists, and
2. the concavity of f is opposite on opposite sides of x_0.

If f has an inflection point at x_0 and f''(x_0) exists, then f''(x_0) = 0

The second derivative test

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