The test states: If the function is twice differentiable in a neighborhood of a stationary point , meaning that , then:
In the last case, the function may have a local maximum or minimum there, but the function is sufficiently "flat" that this is undetected by the second derivative. Such an example is .
Suppose we have (the proof for is analogous). Then
Thus, for h sufficiently small we get
Now, by the first derivative test we know that has a local minimum at .
The second derivative test may also be used to determine the concavity of a function as well as a function's points of inflection. First, all points at which are found. In each of the intervals created, is then evaluated at a single point. For the intervals where the evaluated value of the function is concave down, and for all intervals between critical points where the evaluated value of the function is concave up. The points that separate intervals of opposing concavity are points of inflection.
Second Derivative of the Finger Arterial Pressure Waveform: An Insight into Dynamics of the Peripheral Arterial Pressure Pulse
Sep 01, 2005; Summary The study investigated second derivative of the finger arterial pressure waveform (SDFAP) in 120 healthy middle-aged...