Can an interval with all positive functions of x be proven?

In summary, the interval (a.b) is an open interval that is centered at c and f(x)>0 for all x that are elements of (a,b).
  • #1
mathkillsalot
25
0
Can an interval with all positive functions of x be proven??

f(x) has to always be greater that 0. How do you prove this??

f is continuous as c and f(c)>0. prove that there is an open interval (a.b) centered at c such that f(x)>0 for all x that are elements of (a,b)
 
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  • #2


This comes straight from the definition of continuity. Apply the definition of continuity to your point c. What does it say?
 
  • #3


lim f(x) = f(c)
x -> c ?

but i don't get how it proves that f(a) and f(b) are positive...
 
  • #4


There's another definition of continuity. Use the epsilon-delta definition and choose an appropriate value for epsilon.
 
  • #5


wait...so since f(c) is positive, and (a,b) are approaching c, it means that f(a) and f(b) are approaching a positive number f(c)? and that they are positive?
 
  • #6


mathkillsalot said:
wait...so since f(c) is positive, and (a,b) are approaching c, it means that f(a) and f(b) are approaching a positive number f(c)? and that they are positive?

I'm not sure what you mean by (a,b) approaching c, but it sounds like you do have the right idea. Since f is continuous at c, we can choose values that are close enough to c (a and b) such that f(a) and f(b) are both positive and f is positive on (a,b). Unfortunately, your proof needs to be a little more rigorous than this. From here, you need to use the epsilon-delta definition of continuity to actually prove it.
 
  • #7


okaaay...thankyou :))
and I forgot something, we're supposed to apply the sign preserving property on this. Do i still need the epsilon-delta definition of continuity?
can't I just assign positive values for x?? or is that wrong?
 
  • #8


If you can use the sign-preserving property, it's trivial. The answer follows straight from the definition and you don't need the e/d definition.
 
  • #9


okay, thank you so much :)))))))
 
  • #10


Essentially the same question (about proving the "sign preserving property") was asked and answered here-
https://www.physicsforums.com/showthread.php?t=460196

By the way, "mathkillsalot" you might want to change that user-name. The very people who would help you the most are likely to take exception to it.
 
  • #11


thankyou :))
and I just requested (chroot?) to change my username... hahaha
 

FAQ: Can an interval with all positive functions of x be proven?

Can you provide an example of an interval with all positive functions of x?

Yes, an example of an interval with all positive functions of x is the interval (0,∞), which includes all positive real numbers. This interval can be proven to have all positive functions of x by showing that any value of x within this interval will result in a positive function output.

How can you prove that an interval has all positive functions of x?

To prove that an interval has all positive functions of x, you can use the Intermediate Value Theorem. This theorem states that if a function is continuous on a closed interval, and the function takes on two different values at the endpoints of the interval, then it must also take on every value between those two endpoints. By showing that the function is always positive within the interval, you can prove that the interval has all positive functions of x.

Is it possible for an interval to have both positive and negative functions of x?

Yes, it is possible for an interval to have both positive and negative functions of x. An example of this is the interval (-1,1), which includes both positive and negative real numbers. This interval cannot be proven to have all positive functions of x, as there are values of x within the interval that will result in negative function outputs.

What is the significance of an interval having all positive functions of x?

An interval having all positive functions of x is significant because it means that the function will always output positive values within that interval. This can be useful in many applications, such as in physics or finance, where negative values may not make sense. It also allows for easier analysis and understanding of the function within that interval.

Are there any limitations to proving that an interval has all positive functions of x?

Yes, there are limitations to proving that an interval has all positive functions of x. One limitation is that the function must be continuous within the interval. If the function is not continuous, the Intermediate Value Theorem cannot be used to prove that the interval has all positive functions of x. Additionally, the function must be defined for all values of x within the interval in order for it to have all positive functions of x.

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