Numerical experimentation seems to suggest that $$f_8(x) = \sqrt{1} - \sqrt{\binom{x}{1}} + \sqrt{\binom{x}{2}} - \sqrt{\binom{x}{3}} + \sqrt{\binom{x}{4}} - \sqrt{\binom{x}{5}} + \sqrt{\binom{x}{6}} - \sqrt{\binom{x}{7}} + \sqrt{\binom{x}{8}} \ge 0$$ whenever $x \ge 7$. In general, $$f_{2n}(x) = \sum_{k=0}^{2n} (-1)^k\sqrt{\binom{x}{k}} \ge 0$$ whenever $x \ge 2n-1$. (Here $\binom{x}{k}$, for $x$ real and $k$ a nonnegative integer, denotes the generalized binomial coefficient $\frac{x(x-1)(x-2)\cdots (x-k+1)}{k!}$.)
Looking at the plot of $f_8(x)$ suggests that $$f_{2n}'(x) \ge 0$$ for all $x \ge 2n-1$.
Any ideas (or even better, full solutions) are appreciated.