Solving Proofs similar to AM-GM inequality (1 Viewer)

[Chiprr]

I've generally solved the AM-GM inequality questions by working from what I have to the known statement of (a-b)^2>=0, but the more I look at solutions and from watching mcgrathematics they usually solve them from the known statement to what is given.

Is it wrong for me to be working toward a known statement or should I be working from a known statement to what is given?

 

[Re_x_ources]

Well, no, technically it is not wrong as long as you either use the logical equivalence symbol (<=>) in all your steps or you rewrite your entire thing the right way around and state "via backwards reasoning".

Generally, you are supposed to solve from a known statement to the given, as, if you do it via your current method (i.e. a+b>sqrt(2ab) = (a-2sqrt(ab)+b) = (sqrt(a)-sqrt(b))^2>=0), it is kinda invalid as you used the information you were given first.

You generally don't want to use the information you are given to prove something (i.e. Oh, because the question says the vectors are perpendicular, therefore, the dot product is 0, therefore, the vectors are perpendicular is just an extreme example of circular reasoning), but you generally want to start from a known statement.

But hey, how would you know what the known statement is?

a) Practice enough questions that you know how to prove the 2nd case Cauchy Inequality (otherwise known as AM-GM inequality) by just starting from (sqrt(a)-sqrt(b))^2>=0.

b) Or, do backwards reasoning. (a.k.a. Do your original method, then rewrite it in reverse (as if you started with (sqrt(a)-sqrt(b))^2>=0.), and rub out your original method).

Hopefully this is clear enough, otherwise, feel free to ask me to elaborate!

 

[Chiprr]

Well, no, technically it is not wrong as long as you either use the logical equivalence symbol (<=>) in all your steps or you rewrite your entire thing the right way around and state "via backwards reasoning".

Generally, you are supposed to solve from a known statement to the given, as, if you do it via your current method (i.e. a+b>sqrt(2ab) = (a-2sqrt(ab)+b) = (sqrt(a)-sqrt(b))^2>=0), it is kinda invalid as you used the information you were given first.

You generally don't want to use the information you are given to prove something (i.e. Oh, because the question says the vectors are perpendicular, therefore, the dot product is 0, therefore, the vectors are perpendicular is just an extreme example of circular reasoning), but you generally want to start from a known statement.

But hey, how would you know what the known statement is?

a) Practice enough questions that you know how to prove the 2nd case Cauchy Inequality (otherwise known as AM-GM inequality) by just starting from (sqrt(a)-sqrt(b))^2>=0.

b) Or, do backwards reasoning. (a.k.a. Do your original method, then rewrite it in reverse (as if you started with (sqrt(a)-sqrt(b))^2>=0.), and rub out your original method).

Hopefully this is clear enough, otherwise, feel free to ask me to elaborate!

That makes sense. It'll take a bit of adjusting to and will make some questions feel a little more confusing to approach in the beginning but I'd rather be safe than lose marks. I guess it should've made more sense to go from a known statement to what is given since that is sort of the approach with the triangle inequality in a way I guess (set rule that you apply to a question and follow working through).

I do remember a discussion in class where my teacher looked in the syllabus and told us that we weren't allowed to quote Cauchy's inequality and would need to prove it every time we wanted to use it, so I guess it would've been better to just solve every question from the known statement.

Thanks for the help.

 

[constexpr]

muahahahah, thank the fucking lord i'm done with this bullshit

on a side note tho, scaling on ext 2 was insane. keep at all costs

 

[Luukas.2]

a) Practice enough questions that you know how to prove the 2nd case Cauchy Inequality (otherwise known as AM-GM inequality) by just starting from (sqrt(a)-sqrt(b))^2>=0.

b) Or, do backwards reasoning. (a.k.a. Do your original method, then rewrite it in reverse (as if you started with (sqrt(a)-sqrt(b))^2>=0.), and rub out your original method).

Rather than working from the result to a starting point and then re-writing backwards, you can use proof by contradiction.

For example, if I needed to prove that

​

I could assume that theorem was false, and then proceed

​

The conclusion that I have reached is impossible, because and so is zero or a positive real number.

Since each step following the assumption follows logically, and yet the result is impossible, the logical flaw in the reasoning must lie with the assumption itself... and hence, the assumption itself is false and the theorem must be true. QED.