The mean and variance of $$10$$ observations were calculated as $$15$$ and $$15$$ respectively by a student who took by mistake $$25$$ instead of $$15$$ for one observation. Then, the correct standard deviation is______.

We are given a function $$f: [0, 1] \to \mathbb{R}$$ that is twice differentiable on $$(0, 1)$$ with $$f(0) = 3$$ and $$f(1) = 5$$, and we know the line $$y = 2x + 3$$ intersects the graph of $$f$$ at exactly two distinct points in $$(0, 1)$$. Noting that at $$x = 0$$ we have $$y = 3 = f(0)$$ and at $$x = 1$$ we have $$y = 5 = f(1)$$, the line passes through the endpoints of the graph of $$f$$.

Define $$g(x) = f(x) - (2x + 3)$$. Then $$g(0) = 0$$ and $$g(1) = 0$$, and by hypothesis there are exactly two points $$x_1, x_2 \in (0, 1)$$ where $$g(x) = 0$$.

Since $$g$$ vanishes at four points $$0, x_1, x_2, 1$$, Rolle’s theorem implies the existence of points $$c_1 \in (0, x_1)$$, $$c_2 \in (x_1, x_2)$$, and $$c_3 \in (x_2, 1)$$ such that $$g'(c_1) = g'(c_2) = g'(c_3) = 0$$. Applying Rolle’s theorem again to $$g'$$ on the intervals $$(c_1, c_2)$$ and $$(c_2, c_3)$$ yields points $$d_1 \in (c_1, c_2)$$ and $$d_2 \in (c_2, c_3)$$ where $$g''(d_1) = g''(d_2) = 0$$. Since $$g''(x) = f''(x)$$, it follows that $$f''(d_1) = f''(d_2) = 0$$.

Hence, the minimum number of points in $$(0, 1)$$ where $$f''(x) = 0$$ is $$\boxed{2}$$.