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Derivative test. 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 ...
Second partial derivative test. The Hessian approximates the function at a critical point with a second-degree polynomial. In mathematics, the second partial derivative test is a method in multivariable calculus used to determine if a critical point of a function is a local minimum, maximum or saddle point.
However, not all critical points are extrema. One can often distinguish whether a critical point is a local maximum, a local minimum, or neither by using the first derivative test, second derivative test, or higher-order derivative test, given sufficient differentiability. [5]
Fermat's theorem is central to the calculus method of determining maxima and minima: in one dimension, one can find extrema by simply computing the stationary points (by computing the zeros of the derivative), the non-differentiable points, and the boundary points, and then investigating this set to determine the extrema.
The second derivative of f is the everywhere-continuous 6x, and at x = 0, f″ = 0, and the sign changes about this point. So x = 0 is a point of inflection. More generally, the stationary points of a real valued function f : R n → R {\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} } are those points x 0 where the derivative in every ...
The conditions that distinguish maxima, or minima, from other stationary points are called 'second-order conditions' (see 'Second derivative test'). If a candidate solution satisfies the first-order conditions, then the satisfaction of the second-order conditions as well is sufficient to establish at least local optimality.
Lagrange multiplier. In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variables). [1]
Extreme value theorem. 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.