Functions with one variable

Recall from calculus that the derivative is the rate of change of a single-value function. The first derivative is useful mathematically because points where the first derivative are zero indicate the function s inflection points, which can either be a maxima or a minima. The sign of the second derivative where the first derivative equals zero tells whether the function is maximal (negative second derivative) or minimal (positive second derivative). Hence, necessary conditions for x to be a local minima (maxima) of f(x) on the interval a x b are that f(x) is twice differentiable, i.e., a second derivative exists, 

The term necessary conditions implies that if any of these conditions are not met, then x is not a local minimum (maximum). On the other hand, if these conditions are met, this is no guarantee that x is a global minimum. A function f(x) defined on the interval (a, b attains its global minimum at the point x if and only if f(x ) f(x) for all x in the interval (a, b. In other words, the global minimum is obtained at the smallest value of Y for all x in the interval (a, b.  

When a function is not unimodal, many local minima may be present. The global minimum is then obtained by identifying all local minima and selecting the local minima with the smallest value. The same pro- 

One method to finding the minimum of a function is to search over all possible values of x and find the minimum based on the corresponding values of Y. These algorithms include region elimination methods, such as the Golden Search algorithm or point-estimation methods, such as Powell s method, neither of which require calculation of the derivatives. However, these types of methods often assume unimodality, and in some cases, continuity of the function. The reader is referred to Reklaitis Ravindran, and Ragsdell (1983) or Rao (1996) for details. 

Another method for finding the minimum of a function, one that requires the function to be twice differentiable and that its derivatives can be calculated, is the Newton-Raphson algorithm. The algorithm begins with an initial estimate, xi, of the minimum, x. The goal is to find the value of x where the first derivative equals zero and the second derivative is a positive value. Taking a first-order Taylor series approximation to the first derivative (dY/dx) evaluated at xi 

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