On Apr 10, 1:24 pm, "chapkovski" <chapkov...@gmail.com> wrote:
> how many roots does this equation have?
> Thanks in advance for explaining
For x<= 0 look at the function y=2^x-x^12 which is strictly increasing from -00 (atx= -00) to 1 at x=0 .Therefore there is one root here.
for x>0 try to solve x^12/2^x=1 .take the natural logarithm(ln) to reach the equivalent equation (lnx)/x=(ln2)/2 .The graph of y=(lnx)/x is easily sketched from calculus from which you can read off the number of roots.smn
Two real ones, approximately at x_0 ~= -.9467803304 and at x_1 ~= 1.063346831, given by Lambert's W function as:
x = -12*W((+/-) log(2)/12)/log(2)
If you are looking for complex roots, there are more, given by more complicated exressions in terms of the same function. Briefly, the equation can be solved using Lambert's W function as follows:
> Two real ones, approximately at x_0 ~= -.9467803304 and at x_1 ~= 1.063346831, > given by Lambert's W function as:
> x = -12*W((+/-) log(2)/12)/log(2)
2^x is smaller than x^{12} at x = -1, bigger at x = 0, smaller at x = 2, and bigger at x = 84 ( 2^{84} = (2^7)^{12} = 128^{12} > 84^{12} ), so three real solutions.
-- Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)
> > Two real ones, approximately at x_0 ~= -.9467803304 and at x_1 ~= > > 1.063346831, given by Lambert's W function as:
> > x = -12*W((+/-) log(2)/12)/log(2)
> 2^x is smaller than x^{12} at x = -1, > bigger at x = 0, > smaller at x = 2, > and bigger at x = 84 ( 2^{84} = (2^7)^{12} = 128^{12} > 84^{12} ), > so three real solutions.
Yes indeed. The two roots given by Ioannis are obtained using the principal branch of the Lambert W function. The third root is given by
> > > Two real ones, approximately at x_0 ~= -.9467803304 and at x_1 ~= > > > 1.063346831, given by Lambert's W function as:
> > > x = -12*W((+/-) log(2)/12)/log(2)
> > 2^x is smaller than x^{12} at x = -1, > > bigger at x = 0, > > smaller at x = 2, > > and bigger at x = 84 ( 2^{84} = (2^7)^{12} = 128^{12} > 84^{12} ), > > so three real solutions.
> Yes indeed. The two roots given by Ioannis are obtained using the principal > branch of the Lambert W function. The third root is given by
> x = -12*W_{-1}(-log(2)/12)/log(2)
> where W_{-1} denotes the -1 branch.
Thanks to both for spotting the omission. I did an evalf(") on the exact solutions Maple provided to scan for real solutions, but I missed that one, which seems to be close to x_3 ~= 74.66932553
Moral of the story: When an equation is solvable exactly by W, *always* check the -1 branch as well :-)
> > > > Two real ones, approximately at x_0 ~= -.9467803304 and at x_1 ~= > > > > 1.063346831, given by Lambert's W function as:
> > > > x = -12*W((+/-) log(2)/12)/log(2) > Thanks to both for spotting the omission. I did an evalf(") on the exact > solutions Maple provided to scan for real solutions, but I missed that one, > which seems to be close to x_3 ~= 74.66932553
> Moral of the story: When an equation is solvable exactly by W, *always* check > the -1 branch as well :-)
Not at all. The moral of the story is to not wave great hefty tools like Lambert W functions around when techniques that the average 10 year old should know provide more insight into the solution.
I am aghast that you seemed to think that x^12 would dominate 2^x as x increased.
Phil -- "Home taping is killing big business profits. We left this side blank so you can help." -- Dead Kennedys, written upon the B-side of tapes of /In God We Trust, Inc./.
> > Moral of the story: When an equation is solvable exactly by W, *always* check > > the -1 branch as well :-)
> Not at all. The moral of the story is to not wave great hefty tools > like Lambert W functions around when techniques that the average > 10 year old should know provide more insight into the solution.
Nah. I was just too bored to perform a full analysis of the equation.
> I am aghast that you seemed to think that x^12 would dominate 2^x > as x increased.
It wasn't my fault. Honest :-) That was Maple V release 4's fault. To make sure that I got the right behavior at +oo, I tried a quick:
> plot(x^12-2^x,x=-infinity..infinity);
Unfortunately, version V of Maple misbehaves on this graph, producing slightly different variants each time. The time I checked, it produced a spike at +oo for x->oo (erroneous), so I didn't check further. By breaking the domain of the graph into smaller regions one can get the correct behavior, but I stupidly relied on just that one glance I got from the -infinity..infinity range, which was incorrect.
Shows the dangers of relying too much on CAS for analyses. Maybe we should ban W altogether...
On Tue, 10 Apr 2007, chapkovski wrote: > how many roots does this equation have?
> Thanks in advance for explaining
Three real roots: look at the (real-variable) equivalent equation
x^2 - 2^(x/6) = 0,
use repeatedly Rolle's Theorem to show that it cannot have more than three roots, and use Intermediate Value Theorem to locate one root at a time:
between (-1) and 0, between 1 and 2, and a large positive one because 2^(x/6) eventually outgrows x^2. (The location is: between 74 and 75).
Non-real roots: there are infinitely many of those because Lambert's W-function has infinitely many branches, and on top of that, the twelfth root function has 12 branches.
My computer equation solver found ten more non-real roots, in the strip -1 < Re(x) < 1. But that is no promise that other roots will stay in that strip - actually there is a root close to 121.55+111.31i.
Read about Lambert's W-function. The smaller of the positive roots was found to be (log means the natural logarithm)
> If you are looking for complex roots, there are more, given by more > complicated exressions in terms of the same function. Briefly, the equation > can be solved using Lambert's W function as follows:
> > I am aghast that you seemed to think that x^12 would dominate 2^x > > as x increased.
> It wasn't my fault. Honest :-) That was Maple V release 4's fault. To make > sure that I got the right behavior at +oo, I tried a quick:
> > plot(x^12-2^x,x=-infinity..infinity);
> Unfortunately, version V of Maple misbehaves on this graph, producing slightly > different variants each time. The time I checked, it produced a spike at +oo > for x->oo (erroneous), so I didn't check further. By breaking the domain of > the graph into smaller regions one can get the correct behavior, but I > stupidly relied on just that one glance I got from the -infinity..infinity > range, which was incorrect.
And indeed, to further my defense for being a careless putz (8*(, here are the two graphs:
The Maple V graph is clearly wrong. It's completely missing the third root. That's the one I saw first and didn't even think twice about exp(x) > x^n, which is an immediate givaway.
The Maple 9 graph also seems to be wrong, although "less wrong" than the Maple V graph. I didn't check this one when I answered the question, but to me it implies that the function reaches +infinity between the second and third root, and that the third root is very large in magnitude, almost close to +infinity.
