A discrete function and its rate of oscillation

Question

Consider a function

$y[n]= \cos[w n ]$, where $n$ is an integer.

I have to prove that this signal will have highest rate of oscillation at $w = \pi$. I was thinking I can take the derivatives (of rate of oscillation) and set condition of maxima, but the variable is a discrete variable and not a continuous one.

How do I actually prove this?

Update

This is what I observe:

at $w = 0$ all samples will have value 1 and at $w = \pi$ samples will have positive and negative values in succession- highest rate of oscillation

Please [see this, page number 16], but they have not proved it.

Here is the corresponding page from the book (Digital Signal Processing: Principles, Algorithms and Applications (3rd Edition) ):

As can be seen they have just verified, and not proved. I want to prove this.

Answer 1

What do you mean by "maximum oscillation"? If the range of $y$, it will be $(-1,1)$ for almost any choice of $w$. If by a local maximum, it isn't a function of $w$, but of $n$. If $\theta=0, y[n]$ will always be $0$ for $w=\pi$, certainly not a maximum oscillation.