proof (1 Viewer)

[onebytwo]

show that pi^e < e^pi

this seems a weird question since we know its true already, its like saying show
1+1 < 5

can any help show a legitimate proof, thanks

 

[who_loves_maths]

Hi onebytwo,

Question 1:

I think this is a celebrated inequality, so am certain that there are probably many different proofs of this out there on the net waiting to be looked up (though I've not seen any myself), but I just had a go at it and here's how I approached it, adhereing to only 4unit methods:

Consider the function: y = x/lnx __________ [continuous] for all x > 1

We know that x dominates lnx as x -> infinity - i.e. lim(x -> infinity) y = infinity

Consider: dy/dx = 1/lnx - 1/ln²x

let y' = 0 to find the minimum (note that any single extremum found must be a minimum for any x > 1 because lim(x -> infinity) y = infinity, so there's no need to perform further derivative tests):

0 = 1/lnx - 1/ln²x

---> lnx = 1 __________ [how convenient!]
---> x = e

Therefore, x = e > 1 gives a singular extremum and so must be a minimum.

Hence, since x = pi > 1, then:

y(pi) = pi/ln(pi) > e

---> pi > e.ln(pi)

Raise both sides to 'e' to obtain:

e^pi > pi^e __________ as required.


Question 2:

Proving that 1+1 < 5 is not easy if you are looking for a mathematically rigorous proof (much harder than the above proof for the e's and pi's thing).

You can obtain a proof through 1) the order axioms of the field of real numbers, with the natural numbers embedded, along with 2) the definitions and properties of the set of natural numbers, N, as [the smallest] inductive set.

i.e. the approach would be axiomatic and you will probably need to know some elementary set theory in order to execute a real proof of this inequality.


Hope this helps .

 

[Templar]

Here's two ways of proving 1+1=5. Brief synopsis only.

1. Proof by Peano's axiom
1+1=1'=2
2+1=2'=3 etc
Still need to know a few basic axioms on fields.

2. Proof by Zermelo Fraenkel set axioms.
I'm not even going to go there.

 

[onebytwo]

thanks for the proof, makes sense now.
for the 1+1<5, i didnt want it to be proved, i was just trying to use an example of why the first inequality should be different to this one

 

[acmilan]

I recall this question, it was in the 2 unit 2004 HSC!