Галерея 3236971
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Галерея 3236971
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What is the Supremum of the above set?
I consider the function f ( x ) = x 1 x , and show that f ( x ) is maximum when x = e .
But here the domain of the set is N . So how can I find the supremum of the above set?
Please anyone help me solve it. Thanks in advance.
You know that as a real function, x 1 / x is increasing on ( 0 , e ) and decreasing on ( e , ∞ ) . The same must be true if we consider it a function on the integers. That means that the maximum among integers must be at either 2 or at 3 (since any other integer input must give a function value strictly smaller than one of these two). Now just check those two input values, and you're done.
Note that 2 1 / 2 = 4 1 / 4 . Given what calculus tells you, you know the supremum is attained at 3 .
Try integers either side of e - as the local maximum over the reals is in that area, the maximum over the natural numbers will be close by.
The first few values are 1 , √ 2 = 1.414 … , 3 √ 3 = 1.442 … , √ 2 , 5 √ 5 = 1.379 … , and we suspect that 3 √ 3 is the supremum (or in fact even maximum) of the sequence.
It only remains to show that for larger n , we have n 1 / n < 3 √ 3 or equivalently, n 3 < 3 n . This follows by induction after observing that the left hand side grows by a factor of ( n + 1 ) 3 n 3 = 1 + 3 n + 3 n 3 + 1 n 3 ≤ 1 + 1 + 1 3 + 1 27 < 3 for n ≥ 3 .
Let f ( x ) = x 1 x than ln f ( x ) = 1 x ln x and f ′ ( x ) f ( x ) = 1 x 2 − 1 x 2 ln x
f ′ ( x ) = x 1 x x 2 ( 1 − ln x ) < 0 for x ≥ 3 . This means our function is decreasing and we only need to check 3 values:
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RYCROSS INVESTMENTS, INC. is a California Stock Corporation - Ca - General filed on November 3, 2009. The company's filing status is listed as Suspended - Ftb and its File Number is 3236971 . The Registered Agent on file for this company is Antonio Mendoza Carrillo and is located at 15258 Dakota Street, Victorville, CA 92394.
#NamePos.Ht.Wt.2JaMarcus Russell QB6-62558Daunte Culpepper QB6-42659Shane Lechler...
STARR DESIGNS is a Pennsylvania Domestic Fictitious Names filed on July 19, 2004. The company's filing status is listed as Active and its File Number is 3236971 . The company's principal address is 9532 Kistler Valley Rd, Kempton, Berks 19529. The company has 1 contact on record. The contact is Barbara Starr. There are no reviews yet for this ...
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