Posers and Puzzles
31 Oct 05
01 Nov 05
1 edit
Originally posted by ark13x = 1/e is the solution.
f(x)=x^x
What positive value of x minimizes the function? Just an interesting thing I discovered...
used log differentiation to get f'(x) = (1 + ln(x)) x^x. Set this to 0, and x^x cannot be 0, so (1+ln(x)) must be 0, so x = 1/e. I am not going to test for concavity, because it is pretty clear the x^x is concave up.
Is that a satisfactory answer for you, ark?
01 Nov 05
Originally posted by rheymansCorrect! Great job. 🙂
x = 1/e is the solution.
used log differentiation to get f'(x) = (1 + ln(x)) x^x. Set this to 0, and x^x cannot be 0, so (1+ln(x)) must be 0, so x = 1/e. I am not going to test for concavity, because it is pretty clear the x^x is concave up.
Is that a satisfactory answer for you, ark?
What about the positive value of x (in terms of e) which minimizes f(x)=x^(x^x) ? This one I don't know the answer to yet.
01 Nov 05
Originally posted by ark13I just checked it out. That function increases monotonically, and the limit of the function as you approach 0 is 0, so there is no actual minimum. You can say 0 is the greatest lower bound for this function, however.
Correct! Great job. 🙂
What about the positive value of x (in terms of e) which minimizes f(x)=x^(x^x) ? This one I don't know the answer to yet.
01 Nov 05
Originally posted by PBE6f(0) doesn't exist. Neither does f(x) for any negative non-integer value (and x=-0.2).
I just checked it out. That function increases monotonically, and the limit of the function as you approach 0 is 0, so there is no actual minimum. You can say 0 is the greatest lower bound for this function, however.
The limit as x approaches zero from the positive side (there is no limit approaching from the negative side) is 1.
The lower bound of the function is in fact -1.379 (at x=-0.2). The lower bound in the positive domain is 1/e.
01 Nov 05
Originally posted by XanthosNZYou're checking out the wrong function, XanthosNZ. ark13 is asking about x^(x^x), not (x^x)^x. x^(x^x) tends to 0 as x approaches 0 from the positive side, whereas (x^x)^x tends to 1 as you stated. ark13 is also asking about the positive value(s) that minimize said function. Although x=-0.2 does give a value less than 0 for both functions, it does not minimize either function, as any odd negative integer for x will give you a more negative value. But you are right when you say f(0) does not exist (for either function).
f(0) doesn't exist. Neither does f(x) for any negative non-integer value (and x=-0.2).
The limit as x approaches zero from the positive side (there is no limit approaching from the negative side) is 1.
The lower bound of the function is in fact -1.379 (at x=-0.2). The lower bound in the positive domain is 1/e.
01 Nov 05
1 edit
Whoops, sorry I meant f(x)=(x^x)^x 😳 Sorry about that. And yes, I did want the positive value of x.
My calculations gave the positive x min at approx. 0.60653176 and the value at .83198595
I was just curious if anyone could use logarithms to find the x value in terms of e as was done on the previous problem.
02 Nov 05
Originally posted by ark13Hmm, it looks like I was checking out the wrong function. Curses!!
Whoops, sorry I meant f(x)=(x^x)^x 😳 Sorry about that. And yes, I did want the positive value of x.
My calculations gave the positive x min at approx. 0.60653176 and the value at .83198595
I was just curious if anyone could use logarithms to find the x value in terms of e as was done on the previous problem.
02 Nov 05
Originally posted by PBE6Yep, my bad. It was what I said, but not what I meant to say. And although quoting Dr. Seuss will only confuse people further, as it's contrary to what I just said, I will anyway. "I meant what I said, and I said what I meant."
Hmm, it looks like I was checking out the wrong function. Curses!!
02 Nov 05
Originally posted by ark13So, your function is x^(x^2) then. I only get one min at 1/(sqrt(e)).
Whoops, sorry I meant f(x)=(x^x)^x 😳 Sorry about that. And yes, I did want the positive value of x.
My calculations gave the positive x min at approx. 0.60653176 and the value at .83198595
I was just curious if anyone could use logarithms to find the x value in terms of e as was done on the previous problem.
02 Nov 05
Originally posted by rheymansFrom these results, we can assume that the positive minimizing value of x in x^x^x^x is 1/cubrt (e), and etc. Almost makes you wonder why names of operations stop at exponents. Why isn't there an operation that is the number of times a number is put to it's own power? Well, I guess they had to stop somewhere.
😀