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.