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研究一元函数的最大值与最小值,一般只要求出极大值或极小值,问题就可以解决。特别是遇到只有一个逗留点的情形,问题更加简单。事实上,我们有这样的命题: 若函数y=f(x)在区间(有限或无穷,开或闭)上可微;又在(a,b)内有唯一的逗留点C;如果C是极大(小)点,则C就是f(x)在上的最大(小)点。这命题的几何意义是明显的,证明也不难。因此自然会这样猜想: 如果二元函数z=f(x,y)在区域G上是可微的,又在这区域内只有一个逗留点(x_0,y_0),而且这逗留点是极大(小)点,那么,(x_0,y_0)也一定就是f(x,y)在G上的最大(小)点。这猜想是错的。如果区域G不是闭的,固然不能保证最大(小)值存在,即使已知最大(小)值存在,也不一定就在唯一的逗留点上取得。这和一元函数迥然不同。
To study the maximum and minimum values of a one-way function, the problem can be solved as long as the maximum or minimum value is found. Especially in the case of only one stay, the problem is even simpler. In fact, we have the following proposition: If the function y=f(x) is different in the interval (finite or infinite, on or off); there is only one stay point in (a,b) C; If C is a maximal (small) point, then C is the maximum (small) point of f(x) over . The geometric meaning of this proposition is obvious and proof is not difficult. It is therefore natural to guess this: If the binary function z = f(x, y) is subdifferentiable in region G, then there is only one dwelling point (x_0, y_0) in this region, and this dwell point is extremely large ( Small) Then, (x_0, y_0) must also be the largest (small) point of f(x,y) on G. This conjecture is wrong. If the area G is not closed, it cannot be guaranteed that the maximum (small) value exists. Even if the maximum (small) value is known to exist, it is not necessarily obtained at the only point of stay. This is very different from the one-way function.