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How to tell whether a function is positive or negative? 
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#1
May1006, 07:59 AM

P: 10

Hello everbody
I have functions of two variables and I need to determine if they are positive or negative. I am just wondering if anyone can tell me what the best way to do that is? Thanks a lot in advance 


#3
May1006, 10:49 AM

P: 10

f(x,y)=x^2 + sin(y)  x y  4y^3



#4
May1006, 10:52 AM

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P: 9,396

How to tell whether a function is positive or negative?
That function is neither positive nor negative. Perhaps you mean 'where is it positive and where is it negative?'



#5
May1006, 02:27 PM

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#6
May1006, 02:33 PM

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#7
May1006, 02:38 PM

P: 10




#8
May1006, 02:41 PM

Sci Advisor
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P: 9,396

By looking at it and thinking. After all you have complete freedom here to let x or y be absolutely anything.



#9
May1006, 02:42 PM

P: 10

You mean I just need to put random numbers into the equation for both x and y. But that would only work if the function is neither positive nor negative.



#10
May1006, 03:20 PM

P: 837

It might take you awhile if you just chose a bunch of random numbers. What conditions make f(x,y) positive and what makes it negative?



#11
May1006, 03:26 PM

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#12
May1006, 04:30 PM

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#13
May1006, 04:38 PM

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#14
May1006, 05:15 PM

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You do it by 'doing it', there is no universal method. Look at the function. Is it possible to make it negative? Is it possible to make it positive? Fix x, or y, whatever is needed. For a given x what is the minimal value as a function of y? what is the maximal value, etc.



#15
May1106, 03:03 AM

P: 1,157

You could plot the function; see below.
Looks to be negative for x>0, for all y. I think you'll have to look at which terms dominate for, eg. x>0 y>0, x>0 y<0 etc... (and remember that sin takes values between 1 and 1) It looks like the 4y^3 is dominating... (the sin(y) term just superimposes a small oscillation on this curve) 


#16
May1106, 02:56 PM

P: 10

Sorry, I could not open the attached file. And the graph would be a surface, so I can not tell anything from this.



#17
May1106, 06:29 PM

P: 10

I have put three different values for x (20,0,20), so I got a function of y. After that I plotted the new function and the graph shows that it is positive and negative. After that, I put the same values in the original function but now for y and I plotted the new fucntion for x, and that shows that the function is always positive. So, can I now say the function is neither positive nor negative? 


#18
May1106, 08:45 PM

P: 81

Look for discontinuities in the function, and zeros of the function. If there are none, test a point, and the sign of the function at that point will be the sign of the function everywhere. If there are discontinuities or zeros, they will form the boundary between regions. Test each of the regions, and if each test point has the same sign, that is the sign of the function. Something else you can do is take the absolute value of the function. If f = f over the entire domain, then f is positive. If f = f over the entire domain, then f is negative. Otherwise, it's neither. 


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