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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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