Disproving Homework Statement | Homework Equations | Attempt at Solution

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Homework Help Overview

The discussion revolves around a problem set involving the comparison of two functions, specifically addressing the relationship between a linear function and another function that is not polynomial. Participants are examining the implications of constants and growth rates in the context of limits as n approaches infinity.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the idea of whether a constant c can be chosen such that one function consistently dominates another as n increases. There is a focus on the implications of fixing c before determining n and the resulting contradictions that may arise.

Discussion Status

The discussion is actively exploring the relationships between the functions and the implications of their growth rates. Some participants have provided insights into the behavior of the functions at large values of n, suggesting that there is no fixed c that can satisfy the inequality for all n.

Contextual Notes

There is an emphasis on the nature of the functions involved, particularly the distinction between polynomial and non-polynomial functions, which is central to the discussion. The participants are also considering the limits of n and the implications of choosing constants in their reasoning.

Melodia
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Homework Statement



Thanks for the help on the last problem. Here is the final problem set I'm stuck on:


SYPzV.jpg


Homework Equations





The Attempt at a Solution



To me it seems that there will always be a positive c so that cg(n) is greater or equal to f(n). No matter how large n is, since there's no limit to how large c can be (can even be a decimal), wouldn't that always be possible?

Thanks.
 
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First off, g is not a polynomial, as it is not the sum of multiples of integral powers of n. It might be helpful for you to graph the two functions, because you would see that for n larger than about 20, the linear function dominates the other function. Although f(n) = 2n + 3 is a linear function and grows at only a constant rate, that rate is larger than that of the other function, for large enough n.
 
Oooh I see. Since c is fixed and chosen before the n, then there will always be an n that contradicts the statement right?

I'm guessing the same thing applies to this other problem right?

8cBSM.jpg
 
Last edited:
Yes, essentially. For the first problem, there's no fixed value of c that works because you can always make n large enough so that the inequality doesn't hold. In other words, no such c exists.
 

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