Author |
Topic: Monotone descreasing function (Read 388 times) |
|
JP05
Guest
|
Let f: R -> R be a nonzero function with f'''(x) = f(x) and f(0) + f'(0) + f''(0) = 0.
Show that g(x) = |f(x)| + |f'(x)| + |f''(x)| is a monotone decreasing function.
|
|
 IP Logged |
|
|
|
Icarus
wu::riddles Moderator
Uberpuzzler
Boldly going where even angels fear to tread.
Gender: 
Posts: 4863
|
 |
Re: Monotone descreasing function
« Reply #1 on: May 5th, 2006, 3:44pm » |
 Quote  Modify
|
An uninspired solution to this is to simply solve the differential equation (easier to do if you allow complex numbers, then restrict the final answer to be real).
The equation has solutions of the form f(x) = Pex + Qeqx + Rerx, where q and r are the primative cube roots of unity:
q = r2 = (-1 + i sqrt(3))/2
r = q2 = (-1 - i sqrt(3))/2
The condition f(0) + f'(0) + f"(0) = 0 simply means that P = 0.
Restricting to real values gives f(x) = Ae-x/2cos(sx + b), where s = sqrt(3)/2, and A and b are arbitrary.
Now, a rather tedious calculation should provide the result.
|
| « Last Edit: May 5th, 2006, 3:49pm by Icarus » |
IP Logged |
"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
|
|
|
Eigenray
wu::riddles Moderator
Uberpuzzler
Gender:
Posts: 1948
|
|
Re: Monotone descreasing function
« Reply #2 on: May 5th, 2006, 11:58pm » |
Quote Modify
|
I hadn't really thought much about this problem until now, when I decided to see just how untedious I could make the calculation:
Let h = f + f' + f''. Then h' = h, and since h(0)=0, h(x)=0 for all x. So there aren't many cases to consider.
For example, if on some interval f and f' are positive, then f'' is negative, and so
g = |f| + |f'| + |f''| = f + f' - f'' = -2f'',
which is decreasing, having derivative -2f < 0. The other cases are similar (or one may replace f by f', -f, etc).
Thus g' < 0 on all but the discrete set of points where one of f, f', f'' is 0, and it follows g is monotone decreasing.
|
|
IP Logged |
|
|
|
Icarus
wu::riddles Moderator
Uberpuzzler
Boldly going where even angels fear to tread.
Gender:
Posts: 4863
|
|
Re: Monotone descreasing function
« Reply #3 on: May 6th, 2006, 2:41pm » |
Quote Modify
|
A much more inspired approach!
After the earlier post, I actually traced the same reasoning out almost to the end, but got bogged down by a bad choice in handling the various possibilities for g. As for my "tedious calculation", you avoided it altogether.
|
|
IP Logged |
"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
|
|
|
JP05
Guest
|
|
Re: Monotone descreasing function
« Reply #4 on: May 20th, 2006, 11:23am » |
Quote Modify
 Remove
|
That's neat!
This problem is one of few that I did not have a solution to. I actually found it difficult to get anywhere with it. I really couldn't figure out what to do with the values at 0 but noting that since f''' = f, then f^(4) = f", f^(5) = f''' = f, and so on which then allows you to form other sums of f^(k)'s equal to 0 as well.
|
|
IP Logged |
|
|
|
|
RIDDLES SITE
WRITE MATH!
Home 
Help 
Search 
Members 
Login 
Register
Reply
Notify of replies
Send Topic
Print