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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 ...
The second derivative test can still be used to analyse critical points by considering the eigenvalues of the Hessian matrix of second partial derivatives of the function at the critical point. If all of the eigenvalues are positive, then the point is a local minimum; if all are negative, it is a local maximum.
In mathematics, the comparison test, sometimes called the direct comparison test to distinguish it from similar related tests (especially the limit comparison test), provides a way of deducing whether an infinite series or an improper integral converges or diverges by comparing the series or integral to one whose convergence properties are known.
The derivative of f ′ (if it has one) is written f ′′ and is called the second derivative of f. Similarly, the derivative of the second derivative, if it exists, is written f ′′′ and is called the third derivative of f. Continuing this process, one can define, if it exists, the n th derivative as the derivative of the (n-1) th ...
Calculus of variations is concerned with variations of functionals, which are small changes in the functional's value due to small changes in the function that is its argument. The first variation [l] is defined as the linear part of the change in the functional, and the second variation [m] is defined as the quadratic part. [22]
One can do this either by evaluating the function at each point and taking the maximum, or by analyzing the derivatives further, using the first derivative test, the second derivative test, or the higher-order derivative test.
An illustration of the five-point stencil in one and two dimensions (top, and bottom, respectively). In numerical analysis, given a square grid in one or two dimensions, the five-point stencil of a point in the grid is a stencil made up of the point itself together with its four "neighbors".
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