If $$R$$ is the least value of $$a$$ such that the function $$f(x) = x^2 + ax + 1$$ is increasing on $$[1, 2]$$ and $$S$$ is the greatest value of $$a$$ such that the function $$f(x) = x^2 + ax + 1$$ is decreasing on $$[1, 2]$$, then the value of $$|R - S|$$ is _________.

We are given the quadratic function $$f(x)=x^{2}+ax+1$$ and we have to study its monotonicity on the closed interval $$[1,\,2]$$.

First, recall the basic calculus fact: a differentiable function is increasing on an interval if and only if its derivative is non-negative throughout that interval, and it is decreasing if and only if its derivative is non-positive throughout that interval.

So we begin by finding the derivative. Using the standard power rule $$\frac{d}{dx}(x^{n})=nx^{n-1}$$ and the rule $$\frac{d}{dx}(kx)=k$$ for a constant $$k$$, we differentiate term by term:

$$f'(x)=\frac{d}{dx}(x^{2})+\frac{d}{dx}(ax)+\frac{d}{dx}(1)=2x+a+0=2x+a.$$

Observe that $$f'(x)=2x+a$$ is a linear function in $$x$$ with positive coefficient $$2$$ in front of $$x$$, so $$f'(x)$$ itself is strictly increasing with respect to $$x$$.

Now we treat the two required situations separately.

First, we want $$f(x)$$ to be increasing on $$[1,2]$$. Because $$f'(x)$$ is increasing, its minimum on the interval $$[1,2]$$ is attained at the left end point $$x=1$$. Therefore, the condition for non-negativity of the derivative on the whole interval is

$$f'(1)\ge 0\quad\Longrightarrow\quad 2\cdot1+a\ge 0\quad\Longrightarrow\quad a\ge -2.$$

We are asked for the least value of $$a$$ that makes the function increasing, so we take the boundary value

$$R=-2.$$

Next, we want $$f(x)$$ to be decreasing on $$[1,2]$$. Again, since $$f'(x)=2x+a$$ is increasing in $$x$$, its maximum on $$[1,2]$$ is attained at the right end point $$x=2$$. For the derivative to be non-positive everywhere in the interval we impose

$$f'(2)\le 0\quad\Longrightarrow\quad 2\cdot2+a\le 0\quad\Longrightarrow\quad a\le -4.$$

We need the greatest value of $$a$$ that fulfils this inequality, hence we choose the boundary value

$$S=-4.$$

Finally, we compute the required absolute difference:

$$|R-S|=\left|-2-(-4)\right|=\left|\,(-2)+4\,\right|=\left|\,2\,\right|=2.$$

Hence, the correct answer is Option 2.