(no subject)
Jul. 23rd, 2007 11:40 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
jackOne of Gerald Duck's question was: Which is greater, pi^e or e^pi? I happened to be doing something related yesterday, and was sufficiently amused to continue here.
Spoilers
SNAPE KILLS DUMBLEDORE! Just kidding.
e^pi is greater for any value of pi.
The other question was, "Given a set of n positive integers which sum to 100, what is their maximum product?"
If you relax the constraint on being integers, and there being an integer number of them, and assume they are all the same, you get "Observe that (100/n)^n is low at n=1 and n=100. Where is the maximum?" and differentiating by n (with product and chain rule and letting n=e^logn), you find, unsurprisingly, the answer is always e.
Of course, for the problem with integers, it suggests a sum of 3s is correct, but it's easier to prove it by showing any change makes it smaller.
But it incidentally solves Gerald Duck's problem -- e^x is inherently a maximum :)
Spoilers
SNAPE KILLS DUMBLEDORE! Just kidding.
e^pi is greater for any value of pi.
The other question was, "Given a set of n positive integers which sum to 100, what is their maximum product?"
If you relax the constraint on being integers, and there being an integer number of them, and assume they are all the same, you get "Observe that (100/n)^n is low at n=1 and n=100. Where is the maximum?" and differentiating by n (with product and chain rule and letting n=e^logn), you find, unsurprisingly, the answer is always e.
Of course, for the problem with integers, it suggests a sum of 3s is correct, but it's easier to prove it by showing any change makes it smaller.
But it incidentally solves Gerald Duck's problem -- e^x is inherently a maximum :)
no subject
Date: 2007-07-23 03:11 pm (UTC)no subject
Date: 2007-07-23 03:59 pm (UTC)More concisely: since 2+2 = 2*2, you can ignore 4 as well, so you're left with just 2s and 3s, and at most two 2s. (We're only after the biggest total, not the number of ways of writing it!)
no subject
Date: 2007-07-23 04:12 pm (UTC)no subject
Date: 2007-07-23 05:05 pm (UTC)(i) Can't have 1s.
(ii) WLOG can't have 4s or bigger.
(iii) Can't have three 2s.
Doesn't give you all both possibilities, but gives you the max.
no subject
Date: 2007-07-23 06:25 pm (UTC)(i) n > 1*(n-1), so 1s are always useless
(ii) pulling out a 2 is at least as good if and only if 2(n-2) >= n, which is true iff n >= 4, therefore no point in using 4 or more
(iii) 2+2+2 = 3+3 but 2*2*2 < 3*3 (which is something I love writing down in a degree-level supervision!), so at most two 2s.