Unfortunately, these "ideas" for the problems are low quality. Problem 2: "use derivatives". How the heck that helps if F does not have to be continuous, let alone differentiable.
Out of curiosity, I've searched it and this article was posted to HN at least four times. Opinions aside, turns out it was merely a footnote in the exam pipeline https://news.ycombinator.com/item?id=4761846
If you look at the solution, you'll see how the idea of a derivative helps in that problem.
In this case, a simpler hint is: "Do not use derivatives or anything fancy really. Only solutions are constant functions. Proof by contradiction. Use binary search".
One may only wonder what level were the problems given to regular candidates.