Can we use differentiation in Real analysis?

[phydis]

I'm a beginner to Real Analysis, My problem is, Can we use differentiation when we have to find Suprimum or Infimum for a given set?

A = {(x)^(1/x) | x in N}
I got Sup(A) = e^(1/e) by using differentiation. is it a correct way to find Sup(A)?
or is there any other way to find Sup(A) ?

Thanks.

 

[mfb]

x is restricted to the natural numbers. You cannot differentiate functions on natural numbers, and e is certainly not a natural number.

You can use the extension of the set to the real numbers (and the derivative there) to show things like $x^{1/x} > (x+1)^{1/(x+1)}$ for x>e.

 

[HallsofIvy]

I'm a beginner to Real Analysis, My problem is, Can we use differentiation when we have to find Suprimum or Infimum for a given set?

A = {(x)^(1/x) | x in N}

If x= 1, x^{1/x}= 1^1= 1.
If x= 2, x^{1/x}= 2^{1/2}= \sqrt{2} or about 1.414
If x= 3, x^{1/x}= 3^{1/3}= \sqrt[3]{3} or about 1.442
If x= 4, x^{1/x}= 4^{1/4}= \sqrt[4]{4}= \sqrt{2} or about 1.414
If x= 5, x^{1/x}= 5^{1/5}= \sqrt[4]{5} or about 1.380
If x= 6, x^{1/x}= 6^{1/6}= \sqrt[6]{6} or about 1.348

Do you see what is happening? What do you think is the maximum of this set? Do you see that because "x in N" the "supremum" is the same as the "maximum"? The infimum will be harder!

I got Sup(A) = e^(1/e) by using differentiation. is it a correct way to find Sup(A)?
or is there any other way to find Sup(A) ?

Thanks.

 

[vanhees71]

One can use differentiation to check, where the function has an extremum, and then it's easy to check where it takes a maximum for natural numbers.

I'd also take the logarithm before, i.e., setting f(x)=x^{1/x}, I'd investigate
g(x)=\ln[f(x)]=\frac{\ln x}{x}.

 

[Ray Vickson]

I'm a beginner to Real Analysis, My problem is, Can we use differentiation when we have to find Suprimum or Infimum for a given set?

A = {(x)^(1/x) | x in N}
I got Sup(A) = e^(1/e) by using differentiation. is it a correct way to find Sup(A)?
or is there any other way to find Sup(A) ?

Thanks.

The strategy does not always work. I think what you are suggesting is that you should find the real-valued maximum and use that. It will not give the sup if the maximizing x is not an integer. Perhaps you think that you can look at the integers immediately above and below the real maximum, to find the integer maximum. Often this works (and it *does* work in this case), but IT CAN FAIL: it is perfectly possible to devise examples where, eg., the real maximum is at x = 2.5 but the integer maximum is at x = 50.

 

[vanhees71]

This I don't understand. Can you really give such an example?

 

[micromass]

This I don't understand. Can you really give such an example?

See

 

[phydis]

Now it's clear that use of differentiation to find Sup(A) or Inf(A) isn't a proper way, then what are the proper steps of finding Sup(A) and Inf(A) rather than assuming natural numbers to x ?

 

[micromass]

Now it's clear that use of differentiation to find Sup(A) or Inf(A) isn't a proper way, then what are the proper steps of finding Sup(A) and Inf(A) rather than assuming natural numbers to x ?

In this case, the strategy proposed by mfb works best. That is, you notice (or prove) that the sequence $n^{1/n}$ is a decreasing sequence from a certain point onwards.