In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point.
Derivative tests can also give information about the concavity of a function.
The usefulness of derivatives to find extrema is proved mathematically by Fermat's theorem of stationary points.
The first-derivative test examines a function's monotonic properties (where the function is increasing or decreasing), focusing on a particular point in its domain.
However, calculus is usually helpful because there are sufficient conditions that guarantee the monotonicity properties above, and these conditions apply to the vast majority of functions one would encounter.
Stated precisely, suppose that f is a real-valued function defined on some open interval containing the point x and suppose further that f is continuous at x.
It is a direct consequence of the way the derivative is defined and its connection to decrease and increase of a function locally, combined with the previous section.
Suppose f is a real-valued function of a real variable defined on some interval containing the critical point a.
Further suppose that f is continuous at a and differentiable on some open interval containing a, except possibly at a itself.
Again, corresponding to the comments in the section on monotonicity properties, note that in the first two cases, the inequality is not required to be strict, while in the third, strict inequality is required.
The first-derivative test is helpful in solving optimization problems in physics, economics, and engineering.
In conjunction with the extreme value theorem, it can be used to find the absolute maximum and minimum of a real-valued function defined on a closed and bounded interval.
In conjunction with other information such as concavity, inflection points, and asymptotes, it can be used to sketch the graph of a function.
[1] If the function f is twice-differentiable at a critical point x (i.e. a point where f′(x) = 0), then: In the last case, Taylor's theorem may sometimes be used to determine the behavior of f near x using higher derivatives.
The higher-order derivative test or general derivative test is able to determine whether a function's critical points are maxima, minima, or points of inflection for a wider variety of functions than the second-order derivative test.
As shown below, the second-derivative test is mathematically identical to the special case of n = 1 in the higher-order derivative test.
Let f be a real-valued, sufficiently differentiable function on an interval
Also let all the derivatives of f at c be zero up to and including the n-th derivative, but with the (n + 1)th derivative being non-zero: There are four possibilities, the first two cases where c is an extremum, the second two where c is a (local) saddle point: Since n must be either odd or even, this analytical test classifies any stationary point of f, so long as a nonzero derivative shows up eventually.
Say we want to perform the general derivative test on the function
To do this, we calculate the derivatives of the function and then evaluate them at the point of interest until the result is nonzero.
For a function of more than one variable, the second-derivative test generalizes to a test based on the eigenvalues of the function's Hessian matrix at the critical point.
In particular, assuming that all second-order partial derivatives of f are continuous on a neighbourhood of a critical point x, then if the eigenvalues of the Hessian at x are all positive, then x is a local minimum.
If the Hessian matrix is singular, then the second-derivative test is inconclusive.