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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 ...
Thus, the second partial derivative test indicates that f(x, y) has saddle points at (0, −1) and (1, −1) and has a local maximum at (,) since = <. At the remaining critical point (0, 0) the second derivative test is insufficient, and one must use higher order tests or other tools to determine the behavior of the function at this point.
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 second derivative of a function f can be used to determine the concavity of the graph of f. [2] A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function.
In the middle of the 18th century, the theory of differential forms was first studied in the simplest case of 1-forms in the plane, i.e. +, where and are functions in the plane. The study of 1-forms and the differentials of functions began with Clairaut's papers in 1739 and 1740.
The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule): () ′ = ′ wherever f is positive. ...
Bryan Yeshion Schneps, a 21‑year‑old Temple University student, tried to prevent his attackers from gaining entry. He pressed his hands, his shoulders, his knees, his feet, the full weight of his 6'1", 180‑pound body against the door. But his stamina wore thin, and the door swung free. Bryan cried for help.
The complication is that in 1 dimension, one can either move left or right from a point, while in higher dimensions, one can move in many directions. Thus, if the derivative does not vanish, one must argue that there is some direction in which the function increases – and thus in the opposite direction the function decreases. This is the only ...
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