Absolute Maxima and Minima
Absolute maxima and minima refer to the highest and lowest values of a function over a given interval or on a specific domain. These points represent the peak and trough of the function and are important in optimization problems and in understanding the behavior of functions. They are found by analyzing the critical points and endpoints of the function.
3 Key excerpts on "Absolute Maxima and Minima"
eBook - ePub- Gregory Hill, Mel Friedman(Authors)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...Critical points are the first step in the process, since local maxima and minima become candidates for absolute extremes. The only other candidates are the endpoints of the domain. To determine an absolute extreme for a function, compare the function values at the critical points to the function values at endpoints. Absolute (Global) Extremes A function g has an absolute maximum at a point c of its domain if g (c) ≥ g (x) for all x in the domain of g. A function g has an absolute minimum at a point c of its domain if g (c) ≤ g (x) for all x in the domain of g. Figure 5.1 can now be discussed a little more completely. In addition to local extremes, absolute extremes can also be identified. The absolute maximum is h (b) and the absolute minimum is h (c). It should also be noted that even though they are absolute extremes, both are still listed as local extremes. EXAMPLE 5.2 Find the absolute maximum and minimum values for h (x) = x 2 – 2 x – 3 on the closed interval [0, 3]. SOLUTION Find the critical points by solving h ′(x) = 0. Compare this result to the endpoints of the closed interval. h (0) = –3 and h (3) = 0 The absolute minimum value is –4 when x = 1. The absolute maximum value is 0 when x = 3. Not all functions have extremes. Among the infinite assortment of functions, any combination of extremes, local, absolute or no extreme at all—can exist. Consider the graph of y = x 3...
- Stu Schwartz(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...at x = 1 So −2.9 must be an under-approximation to f (0.9). Absolute Extrema Overview: Functions whose domain is all real numbers may have several relative extrema, but it is possible that they have none. An example is f (x) = x 3 which has a critical point at x = 0, but this critical point is neither a relative minimum nor a relative maximum. However, once we restrict the domain to a specific interval, a function must have an absolute maximum and an absolute minimum. The absolute maximum (sometimes called global maximum) is the largest value of f (x) on an interval while the absolute minimum (sometimes called global minimum) is the smallest value of f (x) on the interval. When you are given a function on a domain that is a specific interval and asked to find its range, you are being asked to find the difference between the absolute maximum value of the function and the smallest. Even if the domain is all real numbers, it is possible that a function can have an absolute maximum or minimum or both. For instance, f (x) = x 2 graphs a parabola and we know that the absolute minimum value of the function is zero. f (x) = sin x graphs a wave and we know that the absolute maximum value of the function is 1 and the absolute minimum is −1, each occurring at an infinite number of points. The general technique to find an absolute maximum or absolute minimum of f (x) on the interval [a, b] is: (a) Find critical values of f (x) (x -values where f ′(x) = 0 or f ′(x) is not defined). (b) Use the first derivative test to locate relative extrema of f (x). (This step is optional). (c) Evaluate f (x) at critical values and at the endpoints...
eBook - ePub- Arsen Melkumian(Author)
- 2012(Publication Date)
- Routledge(Publisher)
...5 Optimization of univariate functions Many problems in economics require identification of an optimal outcome. This task is achieved by maximizing or minimizing some economic entity. For example, a firm may seek to maximize profit π by choosing the optimal level of output Q. Since π is the difference between total revenue (TR) and total cost (TC) we can write where π (Q) is known as the objective function and Q is referred to as the choice variable. The objective function π (Q) is the function that we seek to optimize. Other problems in economics require optimizing across two or more choice variables: one such example would be to minimize a cost function with respect to capital and labor. Another example would be to maximize a profit function with respect to capital, labor and land. In this chapter we will confine ourselves to examining objective functions with one variable, otherwise known in calculus as univariate functions. Finding extreme values of univariate functions is one of the most widely known and applied areas of calculus. 5.1 Local and global extrema In mathematics, maxima and minima are known collectively as extrema. The following is a rigorous definition of local and global extrema. D EFINITION 5.1 Let f (x) be a function defined on an interval I and A = (a, f (a)) be a point on the graph of the function. Then f (x) has: (i) a global maximum at A if f (a) ≥ f (x) for all x ∈ I ; (ii) a global minimum at A if f (a) ≤ f (x) for all x ∈ I ; (iii) a local maximum at A if f (a) ≥ f (x) for all x in a neighborhood of a ; (iv) a local minimum at A if f (a) ≤ f (x) for all x in a neighborhood of a ; (v) a strict global...
