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Process Modelling and Optimization of Hardness in Laser Cladding of Inconel® 625 Powder on AISI 304 Stainless Steel
Published in Samson Jerold Samuel Chelladurai, Suresh Mayilswamy, Arun Seeralan Balakrishnan, S. Gnanasekaran, Green Materials and Advanced Manufacturing Technology, 2020
S. Sivamani, M. Vijayanand, A. Umesh Bala, R. Varahamoorthi
The first derivative test is used to find local extreme points, either maximum or minimum. A local maximum is where the function starts to decrease and a local minimum is where the function starts to increase. Critical points are where a function can have a local maximum or minimum, and are the only places where the first derivative can change sign (Stewart 2012). Let c be a critical point for a continuous function f(x): If f′(x) changes from positive to negative at c, then f(c) is a local maximum.If f′(x) changes from negative to positive at c, then f(c) is a local minimum.If f′(x) does not change sign at c, then f(c) is neither a local maximum or minimum.
A Review of Calculus
Published in Richard L. Shell, Ernest L. Hall, Handbook of Industrial Automation, 2000
The function f is said to have a local minimum at a point x = a if there is a neighborhood of a in which f(x) > f(a). In this case, the value of f(a) is called the local minimum value. It is said to have a global minimum at x = a if f(x) > f(a) for every x in the domain of f. In this case, the value of f(a) is called the global minimum value. For example, if f(x) = x2 then f has a global minimum at x = 0 and this global minimum value is equal to 0. If we set f(x) = (x − 1)(x − 2)(x − 3) and Dom(f) = [0, 5], then f has a local minimum at x = 2 + 1/3 which is not a global minimum since this occurs at x = 0. If f is differentiable we can check the nature of a critical point, a, of f by using the first derivative test for a minimum; that is, if f′(x) < 0 for x in a left neighborhood (Sec. 4.2.2) of a and f(x) > 0 for x in a right neighborhood of a, then f has a local minimum at x = a. In the event that f is twice differentiable on its domain, there is the second derivative test for a minimum which states that if x = a is a critical point of f and f″(a) > 0 then it is a local minimum. The global minimum (and its value) is determined by taking that critical point c where f(c) has the smallest minimum value.
Calculus students’ understanding of the vertex of the quadratic function in relation to the concept of derivative
Published in International Journal of Mathematical Education in Science and Technology, 2018
Annie Burns-Childers, Draga Vidakovic
Not only is the quadratic function an important concept in a college algebra or other introductory mathematics courses, but it is also important in calculus courses. In calculus courses, usually a first year calculus course, students are taught how the derivative function can be used to find critical points, c, of a function. Critical points for f are the interior points c of the domain of f for which or does not exist [13]. Once the critical point is found, algebraic calculations can be used to find f(c), the corresponding y-value. For quadratic functions, (c, f(c)) form the coordinates of the vertex. The vertex can be identified as a local maximum or local minimum by the application of what is often referred to as the first-derivative test (see Figure 1).