I think this is my first proof but constructing it

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In summary, the conversation discusses a problem involving finding the number of real roots for a cubic function and generalizing it to a set of cubic equations. The participant also questions the rigor of their proof and the assumptions made about the continuity and behavior of the cubic function. They suggest incorporating the mean value theorem and limit definition to further strengthen the argument.
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DrummingAtom
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I should say that I've never really try to "prove" many math theorems from my own point of view. Usually, I will just read through a proof and try to grasp the main concept.

I stumbled upon this problem:

Show that y = x3 - 3x + 6 only has one real root.

It got me thinking that this can be generalized to a whole set of x3 functions will have only one real root as long as both of the min/max values are in the positive y of the graph.

y' = 3x2 - 3

Min/max values: x = -1, 1

If I plug these values into the function I get:

y(-1) = 8
y(1) = 4

Which are both positive.

I then took the 2nd derivative to show that the concavity of both min/max values can only possibly go through the x-axis once. Which happens to be when x < -1.

I'm lost on if that information was enough to "show that" for that particular problem. Furthermore, how can I begin generalizing this to a set of x3 equations?

Thanks for any help.
 
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Nice work, the question begged is how rigorous do you need to be.

What are you assuming about the cubic function which makes your proof work? This is what you need to be explicit about. For example you are assuming continuity. (How do you know cubics are continuous?)
I believe you are invoking also the mean value theorem and the limiting behavior of the cubic as x->infinity and x-> - infinity.

Given the limits you can invoke the limit definition, i.e. there exist sufficiently large x and sufficiently small x such that f(x) is negative for one and positive for the other.

That's the sort of stuff you need to fill into increase the rigor of your argument.
 

1. What is the process for constructing a proof?

The process for constructing a proof involves breaking down a problem or statement into smaller, more manageable pieces and then using logical reasoning and evidence to support each step. It also involves clearly defining all terms and assumptions, and making sure each step follows logically from the previous one.

2. How do I know if my proof is correct?

A proof is considered correct if it follows all the rules of logic and is supported by sound evidence. It should also be clear, concise, and easy to follow. It is always a good idea to have someone else review your proof and provide feedback before considering it complete.

3. What are some common mistakes to avoid when constructing a proof?

Some common mistakes to avoid when constructing a proof include using circular reasoning, making unsupported assumptions, and not clearly defining all terms. It is also important to avoid logical fallacies and to make sure each step follows logically from the previous one.

4. How do I improve my proof-writing skills?

One way to improve proof-writing skills is to practice regularly. Start with simpler problems and work your way up to more complex ones. It can also be helpful to read and analyze proofs written by others to gain a better understanding of the structure and logic involved.

5. Can I use diagrams or illustrations in my proof?

Yes, diagrams or illustrations can be useful tools to aid in the understanding and communication of a proof. However, they should always be accompanied by clear explanations and logical reasoning to support their use.

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