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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.
A Reassuring Introduction to Support Vector Machines
Published in Mark Stamp, Introduction to Machine Learning with Applications in Information Security, 2017
Speaking of calculus, we know that the way to solve a maximization problem involving a smooth function of one variable x is to compute its derivative, set the result equal to zero, and solve for x. Such a solution x is known as a stationary point, and it must occur at a (local) peak or valley in the graph of the function. We can then use the second derivative test to determine whether the point corresponds to a (local) maximum or minimum.
Investigation of the effect of computer-supported instruction on students’ achievement on optimization problems
Published in International Journal of Mathematical Education in Science and Technology, 2023
Having obtained the tentative solution to the problem, the class moved on to the executing phase to find the solution algebraically. For this phase, the researcher first raised the question, ‘How can the equation determined in the fourth question in Table 4 be stated in terms of ‘y’ and ‘x’, bearing in mind what ‘y’ and ‘x’ stand for as stated in the answer to the sixth question?’ Employing the Pythagorean theorem, the algebraic representation of the optimization function was constructed as . Afterward, the critical values, which were and , were obtained by working out the equation algebraically. Having concluded that the critical value had no meaning in terms of the context of the problem, the first derivative test was applied to the critical value , and it was concluded that the optimization function had a local minimum at that value, considering the change of the sign of the derivative function of the optimization function.