The “turning” behavior of a graph is called its concavity. The second derivative plays a pivotal role in analyzing a function’s concavity.

Suppose that \(f\) is a function whose derivative \(f^\prime\) exists at every poting in an open interval \(I\). Then \(f\) is concave up on \(I\) if \(f^\prime\) is increasing (and therefore \(f^{\prime\prime}\) is positive) over \(I\). \(f\) is concave down on \(I\) if \(f^\prime\) is decreasing (and therefore \(f^{\prime\prime}\) is negative) over \(I\).

In simpler words, if the rate of change of the rate of change is positive, then the function is turning up, otherwise it’s turning down.

To classify relative extrema using the second derivative, we observe that if \(f\) is differentiable for every \(x\) in an open interval \((a, b)\) and that there is a critical value \(c\) in \((a, b)\) for which \(f^\prime(c) = 0\), then

  • \(f(c)\) is a relative minimum if \(f^{\prime\prime} > 0\).
  • \(f(c)\) is a relative maximum if \(f^{\prime\prime} < 0\).

A point of inflection, is a point across which the direction of a concavity changes. If a function \(f\) has an inflection point, it must occur at a point \(x_0\), where \(f^{\prime\prime}(x_0) = 0\) or \(f^{\prime\prime}\) does not exist. An inflection point is a point where the second derivative goes from positive to negative or viceversa.