Finding region positive/negative regions.

Hi there.

I am given these functions and I want to find out in which areas they are positive or negative.

x^y - 1

|x+y| - |x-y|

so the way i tried the first one is i set the function x^y-1 = c and take c = 0, simplifying i get y log (x) = 0. What do i do now?

Second one i did the same and ended up with

|x+y| = |x-y|

Re: Finding region positive/negative regions.

Quote:

Originally Posted by

**Kuma** Hi there.

I am given these functions and I want to find out in which areas they are positive or negative.

x^y - 1

|x+y| - |x-y|

so the way i tried the first one is i set the function x^y-1 = c and take c = 0, simplifying i get y log (x) = 0. What do i do now?

Second one i did the same and ended up with

|x+y| = |x-y|

Let's analyze the function $\displaystyle f(x)= x^{y}-1$. First we have to exclude the half plane where $\displaystyle x<0$ because here f(x) is complex, i.e. neither positive nor negative. The 'border' which devides the region where f(x) is positive and the region where f(x) is negative is done by the equation...

$\displaystyle x^{y}= 1 \implies y\ \ln x=0$ (1)

... which is satisfied for $\displaystyle x=1$ or $\displaystyle y=0$, so that is...

$\displaystyle f(x)>0\ \text{for}\ 0<x<1\ ,\ y<0\ \text{or}\ x>1\ ,\ y>0$

$\displaystyle f(x)<0\ \text{for}\ 0<x<1\ ,\ y>0\ \text{or}\ x>1\ ,\ y<0$

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$