🚀 go-pugleaf

In article <NqsY8.13902$Kx3.1560@newsread1.prod.itd.earthlink.net>,
Darrell <dr6583@hotmail.com> wrote:
>The two numbers can be represented as x and 12/x, where x>0.  We desire to
>find the least value of their sum, i.e. we need to minimize:
>f(x) = x + 12/x, x>0
>This can be done by the usual method of finding the minimum of f:
>f'(x) = 1 - 12/x^2 = 0

	Does no-one do the AM-GM inequality these days?  For x > 0,
{ x + 12/x } / 2 >= sqrt (x,12/x) == sqrt (12), with equality iff
x == 12/x, ie x == sqrt (12).  So x + 12/x has a minimum of 2.sqrt(12)
[or 4.sqrt(3) if you prefer] at that value of x.  No need for the
sledgehammer of calculus!

>Again... [...]
>We maximize:
>V(r) = pi(r^2)(12-r) = 12pi(r^2)-pi(r^3)

	OK, so for 0 < r < 12, we note that r/2 + r/2 + 12-r is
constant, and so cbrt (r/2 . r/2 , (12-r)) <= (r/2 + r/2 + 12-r)/3
== 4, so r/2 . r/2 , (12-r) <= 4^3 == 64, so that V(r) <= 256 pi,
with equality iff r/2 == 12-r, ie r == 8.

	The standard inequalities are your friends for these sorts
of problems!  [I seem to write this article twice per year -- is
there any point having an education group where people don't learn?!
Harrumph.]

	We used to have a standard interview question -- farmer has
400m of fencing with which to enclose a rectangular field, blah, blah.
Applicants would let the sides be a and b, then get into a mess 'cos
they couldn't do calculus with two unknowns, but eventually they got
there.  There were follow-ups where one side was against a cliff, so
only three sides were needed, etc.  But I usually asked them if they
had known what the answer would be before they "proved" it -- "Oh yes,
it's a square with side 100".  OK, so if one side is 100+h, what's
the other side?  "100 - h."  So the area is?  "100^2 - h^2."  So,
what can you say?  "???"  Well, what do you know about h^2?  "It's
positive?"  So -h^2 is?  "Negative?"  And 10000-h^2?  "Positive?"
NO!  "Ah, less than 10000."  With equality when?  "h == 0 [triumph!]."
Happy days.

-- 
Andy Walker, School of MathSci., Univ. of Nott'm, UK.
anw@maths.nott.ac.uk