- #1
Dschumanji
- 153
- 1
I have recently been studying the following function for fun (math nerd):
f(x) = x^(1/x) where x is an integer greater than 0
I want to prove that the function is decreasing for x > 2. It seems that the straight forward approach to this proof is to use induction. It can easily be shown that 3^(1/3) > 4^(1/4), which becomes my base case. I then assume that the following inequality holds:
(k-1)^(1/(k-1)) > k^(1/k)
Using the above inequality I must show that:
k^(1/k) > (k+1)^(1/(1+k))
I have tried everything in my mathematical toolkit to rewrite the first inequality into something that looks like the second but to no avail. My last attempt involved taking the logarithm to the base k-1 on the left side and the logarithm to the base k on the right side. It seems to work but I run into some technicalities that I can't seem to get around when I try to raise each side to a different base. Someone please help!
f(x) = x^(1/x) where x is an integer greater than 0
I want to prove that the function is decreasing for x > 2. It seems that the straight forward approach to this proof is to use induction. It can easily be shown that 3^(1/3) > 4^(1/4), which becomes my base case. I then assume that the following inequality holds:
(k-1)^(1/(k-1)) > k^(1/k)
Using the above inequality I must show that:
k^(1/k) > (k+1)^(1/(1+k))
I have tried everything in my mathematical toolkit to rewrite the first inequality into something that looks like the second but to no avail. My last attempt involved taking the logarithm to the base k-1 on the left side and the logarithm to the base k on the right side. It seems to work but I run into some technicalities that I can't seem to get around when I try to raise each side to a different base. Someone please help!