Compare 1001^(1/1001) and 1002^(1/1002) - Which is Larger?

[lax1113]

Homework Statement

which value is larger?

1001^(1/1001) or 1002^(1/1002)

Homework Equations

?

The Attempt at a Solution

Honestly, I am pretty stuck on what to do here. We can use calculus to prove this (might be needed?) I cannot use a calculator, have to make it obvious that one is larger than the other. Where would i start with this?

 

[Mark44]

Look at the function x^1/x, and in particular, its derivative. If the derivative is negative, the function is increasing. If positive, the function is increasing. One of these should tell you something about the inequality you are investigating.

Be careful! The derivative of x^1/x is NOT (1/x)x^{1/x - 1}.

 

[lax1113]

So derivative of x^1/x is x^(1/x)*ln(x). I am doing it for all x >1, so that would mean that this function is increasing.

I can also do x+1^(1/(x+1)) and the derivative of that, but where can I go from there? Now I just have two derivatives that are no obvious than the original question...

 

[Mark44]

That's not the derivative.

If x > 0, x = e^{ln x}, right? So x^1/x = (e^{ln x})^1/x = e^{(1/x)*ln x}. Now differentiate.