Inequalities Question (1 Viewer)

[theprofitable95]

Prove that

a/b +b/c +c/a is greater than or equal to 3, where a,b and c are positive numbers.

Just can't figure out how to do this one.

I also have a question about inequalities in general. When writing a proof, do we always have to start from a known statement and then reach the statement, or can we just manipulate the original expression, writing 'required to prove' as we go.

I'll use an easy example question. Prove that a^2 + b^2 >2ab

METHOD 1
R.T.P a^2-2ab+b^2>0
RTP. (a-b)^2>0
since result here is obvious, statement has been proved.

OR in exam conditions do i have to re-write this in reverse:
ie. METHOD 2

it is known that (a-b)^2>0
a^2 - 2ab +b^2>0
a^2 +b^2 > 2ab, thus statement is true.

 

[enigma_1]

Like is this induction? I would use method 1 it seems more logical.

 

[Shadowdude]

You need to start from a known statement, and then prove it from there.

The only variant is you manipulate what you need to get to ________, and then prove __________ from something known - and then show that _________ implies the original statement.

Method 2 only works if each step where you go RTP is reversible.

 

[Carrotsticks]

Prove that

a/b +b/c +c/a is greater than or equal to 3, where a,b and c are positive numbers.

Just can't figure out how to do this one.

For the AM-GM inequality for n=3, let x=a/b, y=b/c and z=c/a.

The result falls out immediately.

 

[theprofitable95]

Wow thank you so much... I didn't even think about the AM-GM inequality.

 

[seanieg89]

For the AM-GM inequality for n=3, let x=a/b, y=b/c and z=c/a.

The result falls out immediately.

I didn't think that students were allowed to state/use AM-GM without proof?

 

[Carrotsticks]

I didn't think that students were allowed to state/use AM-GM without proof?

They aren't, but I can't imagine students being asked a question like this flat out without one or two previous parts.