I want to determine whether the function $f: \mathbb{R} \rightarrow \mathbb{R}$, $f(x) = 2x+[x]+\sin x \cos x$ is increasing or decreasing, where $[\cdot]$ denotes the Greatest Integer Function.
I only know the derivative method for checking whether it's increasing or decreasing. But here how do I find the derivative of this function. We are supposed to answer this without using any graphical devices.
You also know the definition of increasing.
It is straightforward to see that the greatest integer less than or equal to a given integer is constant almost everywhere and jumps at the integers. So its derivative is $$ \frac{\mathrm{d}}{\mathrm{d}x} [x] = 0, x \in \Bbb{R} \smallsetminus \Bbb{Z} $$ and is undefined on the integers.
Then, on $\Bbb{R} \smallsetminus \Bbb{Z}$, $$ \frac{\mathrm{d}}{\mathrm{d}x} \left( 2x + [x] + \sin x \cos x \right) = 2+0+\cos^2 x - \sin^2 x \text{.} $$ Cosine and sine are bounded and we know $\cos^2 x \in [0,1]$ and $\sin^2 x \in [0,1]$. so this derivative is bonded below by $2+0-1 = 1$. Consequently, this derivitive is positive on $\Bbb{R} \smallsetminus \Bbb{Z}$. That is, on each open interval $(n,n+1)$ with $n \in \Bbb{Z}$, $f$ is increasing.
Let $n \in \Bbb{Z}$ and consider the limits of $f(x)$ as $x$ approaches $n$ from the left and right. The continuous terms of $f$ are easy to deal with. The greatest integer function increase from $n-1$ immediately to the left of $n$ to $n$ immediately to the right of $n$. \begin{align*} \lim_{x \rightarrow n^-} f(x) &= 2n + (n-1) + \cos n \sin n \\ \lim_{x \rightarrow n^+} f(x) &= 2n + n + \cos n \sin n \end{align*} So we see that $f$ is increases (by $1$) at integral points of its domain.
Therefore, $f$ is increasing.