Logic for finding the mode of the binomial distribution

Question

In Boas' Mathematical Methods probability chapter, it's stated that for the binomial distribution the most probable value of x is approximately np.
It says that this can be shown by finding the values of x for which

$$f(x+1) > f(x) $$ and $$f(x+1) \leq f(x)$$

where $f(x)=\frac{n!}{(n-x)!x!}p^{x}(1-p)^{n-x}$

but no explanation for why this would give the mean is given. Why is this reasonable?

Answer 1

You get something like: $$f(0)<f(1)<f(2)<\cdots<f(m)\geq f(m+1)>f(m+2)>\cdots>f(n)$$ with $m$ close to $np$.

Here $f(m)$ is evidently the maximal of $f(0),\dots,f(n)$.