寻找最大值 [英] Finding the Maximum

问题描述

如何通过计算机软件（如Mathematica和Matlab）找到以下最大值或上游值： $ \sup\frac {（1 + s）^ {4} +（s + t）^ {4} + t ^ {4}} {1 + s ^ {4} + t ^ {4}} $ ？

取而代之的数值近似，准确的最大值是多少？

谢谢。因为这个问题看起来有点像家庭作业，这里的答案有点像演讲：

• 问问你自己会发生什么函数作为 s 和 t 转到小到大的正值和负值;这将帮助您确定您应该检查的值的范围; Mathematica 和 Matlab 可以帮助您的数据显示出来;


• 在您感兴趣的值范围内绘制函数的图形，培养对其形状的感觉并尝试找出它的最大值;为此 Mathematic Plot3D [] 函数和 Matlab plot（）函数都是有用的;


• 因为这是2个变量的函数，所以你应该考虑绘制它的一些部分，即保存 s （或 t ）常量，然后制作截面函数的二维图;再次，了解一下函数的行为方式;现在您应该能够对 s，t 进行某种搜索>值的函数的最大值，并得到一个可接受的准确结果。

如果这太困难，那么您可以使用 Mathematica 函数 NMaximize [] 。我认为 Matlab 对于内置的符号函数具有相同的功能，并且您必须在数字上进行计算，但函数 findmax 会有帮助。

How to find the following Maximum or supremum by computer software such as Mathematica and Matlab: $\sup\frac{(1+s)^{4}+(s+t)^{4}+t^{4}}{1+s^{4}+t^{4}}$?

Instead of numerical approximation, what is the accurate maximum?

Thanks.

解决方案

Since the question seems a bit like homework, here's an answer that starts a bit like a lecture:

• ask yourself what happens to the function as s and t go to small and to large positive and negative values; this will help you to identify the range of values you should be examining; both Mathematica and Matlab can help your figure this out;
• draw the graph of your function over the range of values of interest, develop a feel for its shape and try to figure out where it has maxima; for this the Mathematic Plot3D[] function and the Matlab plot() function will both be useful;
• since this is a function of 2 variables, you should think about plotting some of its sections, ie hold s (or t) constant, and make a 2D plot of the section function; again, develop some understanding of how the function behaves;
• now you should be able to do some kind of search of the s,t values around the maxima of the function and get an acceptably accurate result.

If this is too difficult then you could use the Mathematica function NMaximize[]. I don't think that Matlab has the same functionality for symbolic functions built-in and you'll have to do the computations numerically but the function findmax will help.

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