| 单词 | maximum |
| 释义 | maximum The highest value a function takes for any value of its arguments. A maximum may be local or global. For example, y = 1 - x2 has a global maximum of y = 1 when x = 0; there is no value of x which gives a higher y. It is a necessary condition for a local maximum of a function y = f(x) that dy/dx = 0; given that dy/dx = 0, it is a sufficient but not necessary condition for a local maximum that d2 y/dx2 < 0. Consider, however, the function z = x3 - 3x. This has a maximum z = 2 at x = -1, but this is only a local maximum, that is, z is higher for x = -1 than for nearby values of x. Sufficiently large positive values of x can make z larger without limit, so that z does not have a global maximum. The maximum of a set of xi, i = 1, 2,…, N, written arg max[x1, x2,…, xN ] is the largest value of any xi. The horizontal axis shows x; the vertical axis shows y = f(x). In Panel 1, y = 1 + 2x - x2 ; dy/dx = 2-2x, so dy/dx = 0 for x = 1. d2 y/dx2 = -2, so this is a global maximum. In Panel 2, y = 1 - 2x + x2 ; dy/dx = -2 + 2x, so dy/dx = 0 when x = 1. d2 y/dx2 = 2, so this is a global mimimum. In Panel 3, y = 1 + x - 2x2 + x3 ; dy/dx = 1 - 4x + 3x2, so dy/dx = 0 for x = 1 and x = 1/3. d2 y/dx2 = - 4 + 6x, thus when x = 1, d2 y/dx2 = 2, so there is a local minimum; but when x = 1/3, d2 y/dx2 =-2, so there is a local maximum. See also: constrained maximum, global maximum, and local maximum |
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