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If D(a, b) = 0 then the point (a, b) could be any of a minimum, maximum, or saddle point (that is, the test is inconclusive). Sometimes other equivalent versions of the test are used. In cases 1 and 2, the requirement that f xx f yy − f xy 2 is positive at ( x , y ) implies that f xx and f yy have the same sign there.
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.
Furthermore, a global maximum (or minimum) either must be a local maximum (or minimum) in the interior of the domain, or must lie on the boundary of the domain. So a method of finding a global maximum (or minimum) is to look at all the local maxima (or minima) in the interior, and also look at the maxima (or minima) of the points on the ...
A continuous function () on the closed interval [,] showing the absolute max (red) and the absolute min (blue).. In calculus, the extreme value theorem states that if a real-valued function is continuous on the closed and bounded interval [,], then must attain a maximum and a minimum, each at least once.
The Lagrange multiplier theorem states that at any local maximum (or minimum) of the function evaluated under the equality constraints, if constraint qualification applies (explained below), then the gradient of the function (at that point) can be expressed as a linear combination of the gradients of the constraints (at that point), with the ...
The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers.
The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting of a parabola to the graph of () at the trial value , having the same slope and curvature as the graph at that point, and then proceeding to the maximum or minimum of that parabola (in higher dimensions, this may also be a saddle point), see below.
Fermat's theorem gives only a necessary condition for extreme function values, as some stationary points are inflection points (not a maximum or minimum). The function's second derivative , if it exists, can sometimes be used to determine whether a stationary point is a maximum or minimum.