How to prove the positivity if this function?

Question

Andrea Strappato 2021-10-12

0
链接

此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1562031-how-to-prove-the-positivity-if-this-function

评论： David Goodmanson 2021-10-28

采纳的回答： David Goodmanson

在 MATLAB Online 中打开

Hi,

I need to prove the positivity of the y:

y = x1 - x2 + sin(x1)

such that:

pi>x1>0, x1>x2, x1,x2 real

The Matlab code is:

x = sym('x', [1 2], 'real');
assume(pi>x(1)>0 & x(1)>x(2));
y = x(1) - x(2) + sin(x(1));

We all know that the sum of "x(1)-x(2)" and "sin(x(1))" ,according to the boundary, is always definite positive.

Checking it in Matlab:

isAlways(y>=0)

I get:

Warning: Unable to prove '0 < x1 - x2 + sin(x1)'. 
> In symengine
In sym/isAlways (line 42)

How can I prove the positivity of y?

The whole function is more bigger then y, so I can't simply say "y looks positive", I need to check it using Matlab.

Any kind of help is really appreciated!

0 个评论
显示 -2更早的评论隐藏 -2更早的评论

请先登录，再进行评论。

请先登录，再回答此问题。

Answer 1

David Goodmanson 2021-10-23

0
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1562031-how-to-prove-the-positivity-if-this-function#answer_814928

在 MATLAB Online 中打开

Hi Andrea,

What might constitute a proof using Matlab is a good question. If the answer to isAlways(y>=0) had come out as 1, would that be a proof? I guess maybe it would if the focus is on learning how to use symbolic variables, but it provides zero insight into the inequality itself. It's just the output from a black box. However, rearranging y > 0 gives

sin(x1) > x2 - x1

By the conditions of the problem, the left hand side is positive due to the angle restrictions and the right hand side is negative. So the inequality is correct.

2 个评论
显示无隐藏无

Andrea Strappato 2021-10-23

Thanks for the answer!

The "real" inequality to check is more complex then this, showed in the topic. Maybe solving an easier inequality could help.

Anyway, I've solved this just replacing a set of value for the symbolic variables and check if it is definite positive.

Best.

David Goodmanson 2021-10-28

Hi Andrea,

checking the inequality with a set of values is of course not a mathematical proof, but with enough sets of values it might be considered a 'proof for all practical purposes'.

请先登录，再进行评论。