Is K Always Positive? Investigating the Positivity of a Quadratic Equation

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In summary, the equation is not always positive, and it needs to be proven in what range of a, b, and c it is positive.
  • #1
kcuk220
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Homework Statement



This is the equation where i, a, b and c are positive integer.
[URL]http://www.icedsolo.com/eq.GIF[/URL]

Homework Equations



[URL]http://www.icedsolo.com/eq.GIF[/URL]

The Attempt at a Solution



The question is, proof whether K is always positive, or it is positive in certain range of a, b, and c. This is urgent for me to have above very difficult question resolved, please help!
 
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  • #2
It is not positive in general. For example, for
[tex]a = 100, b_i = 1 + 2^{-i}, c = 1[tex]
and d any value (since it does not occur in your expression :wink:) the expression is negative.

What do you mean by a "range" of a, b and c. Despite your inconvenient notation, I think that {bi} is a series. It is not unlikely that the allowed range for K to be positive for a and c depends on the values of b (for example, if b is very large in the first n terms and then tends to zero, or the other way around).

Why do you need to prove this, if you don't mind my asking?
 
  • #3
Thanks for replying so quick.

I'm doing a project that compares two time t1 and t2, as the result of experiments with two settings. We want to quantify in what settings t1 is larger than t2 (i.e. setting for t2 leads to time saving. So we assume K = t1 - t2 and define the equation above to quantify it.

Therefore, we are trying to find out in what settings (i.e. value of the variables), t1 is always longer than t2.

I understand what you explained above, but I'm now confused what can I prove about the equation for the report, with the equation :(

Btw, it's no d in this equation, I shouldn't have it added sorry.
 
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  • #4
Well IMO the main problem is that you have infinitely many parameters. If, for example, all the [itex]b_i[/itex] were fixed, you could make a plot with a on the horizontal axis and c on the vertical axis, and in the plot indicate the area where K is positive.
Also, in certain cases you may be able to make useful estimates (for example, if c is very large you can neglect the last term and maybe do some inequalities on the products and sum). For example, you can then rewrite (the approximation sign indicates the missing last term):
[tex]K \approx \left( \prod_i b_i \right)\left( \prod_j b_j - 1 \right) - a \sum_k b_k(b_k - 1)[/tex]
and try to get something out of that.
 
  • #5
Thanks again. I'm sure you have given great hints on it. Let me try to assume c be a large number and use your equation.
 
  • #6
OK, good luck. But please don't trust me too much, I was just thinking out loud. There is no guarantee it will get you anything useful.

So any additional information, for example on the bi that might be relevant is welcome.
 

1. What does it mean to "prove K is always positive"?

Proving that K is always positive means demonstrating that the variable K, in any given scenario or equation, will always have a positive value or result. This can be done through mathematical or scientific analysis.

2. Why is it important to prove that K is always positive?

Proving that K is always positive is important because it can provide valuable information about the behavior of a system or process. It can also help in identifying any potential errors or flaws in a theory or model.

3. What methods can be used to prove that K is always positive?

There are various methods that can be used to prove that K is always positive, depending on the specific scenario. Some common methods include mathematical proofs, statistical analysis, and experimental testing.

4. Can K ever be negative or zero?

No, if K is proven to always be positive, then it cannot have a negative or zero value. This means that in any given situation, K will always have a positive value.

5. How does proving K is always positive impact scientific research?

Proving that K is always positive can have a significant impact on scientific research. It can help in validating theories and models, as well as in developing new methods and technologies. It can also provide a better understanding of natural phenomena and contribute to advancements in various fields of science.

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