How to prove the positivity if this function?
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Andrea Strappato
am 12 Okt. 2021
Kommentiert: David Goodmanson
am 28 Okt. 2021
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!
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David Goodmanson
am 23 Okt. 2021
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 Kommentare
David Goodmanson
am 28 Okt. 2021
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'.
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