## Quick Reference

A test, usually to determine the nature of stationary points of a function, which uses one or more derivatives of the function. If *f*′(*α*)=0, so there is a stationary point at *x*=*α*, the first derivative test considers the signs of *f*′(*x*) at *x*=*α*^{+}, *α*^{−}, i.e. whether the gradient is positive, negative, or zero as *x* approaches the stationary point from above and from below. The various possible combinations are shown below.

The second derivative test considers the sign of *f*″(*x*) at *x*=*α*. If it is positive, the value of *f*′(*x*) is increasing as the function goes through *x*=*α*, which requires the stationary point to be a minimum; if *f*″(*α*) < 0 then it must be a maximum, while if *f*″(*α*)=0 it may be that *x*=*α* is a point of inflection but this is not a sufficient condition. To determine the nature in this case, either the first derivative test may be used, or higher derivatives can be considered until the first non-zero higher derivative is found.

*Subjects:*
Mathematics.