# How to tell whether a function is positive or negative?

by behappy
Tags: function, negative, positive
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 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
 Emeritus Sci Advisor PF Gold P: 9,772 What are these functions? ~H
 P: 10 f(x,y)=x^2 + sin(y) - x y - 4y^3
 Sci Advisor HW Helper 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?'
P: 10
 Quote by matt grime That function is neither positive nor negative. Perhaps you mean 'where is it positive and where is it negative?'
So, can you tell me please how you found out that it is neither positive nor negative. When I have a function of two variables, I just need to determine whether it is postive or negative.
P: 695
 So, can you tell me please how you found out that it is neither positive nor negative.
Find a point at which the function is positive, and then find another point where it is negative.
P: 10
 Quote by Muzza Find a point at which the function is positive, and then find another point where it is negative.
And how can I do that?
 Sci Advisor HW Helper P: 9,396 By looking at it and thinking. After all you have complete freedom here to let x or y be absolutely anything.
 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.
 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?
P: 10
 Quote by FrogPad 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?
What do you mean by conditions here?
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HW Helper
P: 9,396
 Quote by behappy 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.
No, I don't mean that. I mean look at it and see that ignoring x it behaves like -y^3 and ignoring y it behaves like x^2 so picking numbers carefully it is easy to show as a function it is neither positive nor negative.
P: 10
 Quote by matt grime No, I don't mean that. I mean look at it and see that ignoring x it behaves like -y^3 and ignoring y it behaves like x^2 so picking numbers carefully it is easy to show as a function it is neither positive nor negative.
Right, picking numbers is an easy way to show if a function is neither positive nor negative. But what about the other cases like when the fucntion is always positive
 Sci Advisor HW Helper P: 9,396 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.
 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) Attached Thumbnails
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.

 Quote by J77 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)
P: 10
 Quote by matt grime 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.

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?
P: 81
 Quote by behappy 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?
If you get both positive and negative values for the function from various choices of the variables, the function is neither positive (i.e. nowhere negative) nor negative (i.e. nowhere positive). There are regions where it is positive, and regions where it is negative. Since plugging in points only can only demonstrate whether a fuction is neither positive nor negative, here's one way you can see if the function is positive or negative:

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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