How to tell whether a function is positive or negative?

[behappy]

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

 

[Hootenanny]

What are these functions?

~H

 

[behappy]

This is one of them...

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

 

[matt grime]

That function is neither positive nor negative. Perhaps you mean 'where is it positive and where is it negative?'

 

[behappy]

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 positive or negative.

 

[Muzza]

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.

 

[behappy]

Find a point at which the function is positive, and then find another point where it is negative.

And how can I do that?

 

[matt grime]

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

 

[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.

 

[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?

 

[behappy]

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?

 

[matt grime]

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.

 

[behappy]

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 function is always positive

 

[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.

 

[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)

 

[behappy]

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

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)

 

[behappy]

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 function for x, and that shows that the function is always positive. So, can I now say the function is neither positive nor negative?

 

[Nimz]

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 function 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.

 

[behappy]

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.

Thanks a lot for your co-operation,

I am so interested in what you have just said, but how I can take the absolute value of the function if the domain is [-inf,inf] for both x and y. And also, I use Mathematica so I have plotted the function and the result is a surface. Can I tell whether the function is positive or not from that.

Also, I have looked for discontinuities in the function, and zeros of the function but I found non. Now, how can I test a point? What do you mean?

 

[J77]

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

ummm...

Your function defines a surface

If I was you, I'd reread my post and then apply it, and the graph, to the questions/conclusions you pose above.