Tools to show a function is decreasing

Question

I was asked to prove that the following function is decreasing

$f(x)=\left(\left(1-\frac{1}{x}\right)^{x}\right)^{x}$ for every $2<x$

I was wondering what tools do I have except the derivate method?

It seems that there is some trick with this specific functions, as $\left(1-\frac{1}{x}\right)^{x}$ is increasing.. Which also quiet confuses me..

Answer 1

$(1-\frac{1}{x})^x$ does increase a little as $x$ increases but that is not the point, the point would be how much it increases and more specifically if it reaches a point where it is $> 1$ as this is going to be raised to a power and any a number greater than $1$ raised to a positive power will get bigger while any positive number less than $1$ raised to a positive power will get smaller.

Take a look at this graph of $(1-\frac{1}{x})^x$ for $x>2$, as you see it does increase a little bit at first but then you can see the curve flatten out even for smaller numbers and most importantly stays below $1$ for all values.

In fact, an interesting result is that $\lim_{x->\infty}(1-\frac{1}{x})^x=\frac{1}{e}\approx 0.367$ so raise that to the power $x$ and the bigger $x$ bigger the more this will increased as it is a positive number smaller than $1$

Answer 2

As $x$ increases, $\left( 1 - \frac{1}{x} \right)$ decreases for $x>2$. Raise that to a power of $x$, it decreases even more. Raise that to a power of $x$ and it decreases even more.

Here's a graph: