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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 ...
(2nd derivative is 0 at that point.) Unique global maximum at x = e. (See figure at right) x −x: Unique global maximum over the positive real numbers at x = 1/e. x 3 /3 − x: First derivative x 2 − 1 and second derivative 2x. Setting the first derivative to 0 and solving for x gives stationary points at −1 and +1. From the sign of 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.
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 ...
One application of higher-order derivatives is in physics. Suppose that a function represents the position of an object at the time. The first derivative of that function is the velocity of an object with respect to time, the second derivative of the function is the acceleration of an object with respect to time, [29] and the third derivative ...
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.
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.
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]