Second Derivative Test

The second derivative test is a useful calculus tool to determine what exactly a critical point may be.

Per ¹, we define the test as:

Theorem 1 (Second derivative test of extrema.) Let \(f(x)\) be a function with \(f^{\prime}\left(x_0\right)=0\). Then if \(f^{\prime \prime}\left(x_0\right)>0\), the function has a local minimum at \(x=x_0\). If \(f^{\prime \prime}\left(x_0\right)<0\), the function has a local maximum at \(x=x_0\). If \(f^{\prime \prime}\left(x_0\right)=0\), the second derivative test fails (point of inflection).

Footnotes

1. https://mathstat.slu.edu/~may/ExcelCalculus/sec-4-5-SecondDerivativeConcavity.html