Minimum Value Question: You definitely can't use Calculus here!?

Are there any restrictions on x? If no, then there is no "minimum" value because the sums will always approach infinity.

f(x) = |x-1| + |x-1| + |x-2| + |x-3| + |x-5| + |x-8| + ...

If you pick any value of x, you will find that your results are just that far away from the regular Fibonacci numbers.

Let x = 1.7; f(1.7) = 0.7 + 0.7 + 0.3 + 1.3 + 3.3 + 6.3 + 11.3... and if you keep adding to infinity then the sum becomes infinity.

By the way, you *can* use calculus on this function...with some restriction.

Suppose x is a really large positive number. This means that

f'(x) = 1 + 1 + 1 + 1 + 1.... you will have a string of +1s before x and then -1s afterward; since there are infinitely many more numbers *larger* than x then f'(x) --> -∞.

Maybe I'm missing something here. It seems to me this is a diverging series.

The Fibonacci numbers grow without bound. So no matter what x you pick, there are infinitely many Fibonacci numbers greater than x. Not only are they greater than x, they are greater than x + 1. So there are infinitely many Fibonacci numbers F_n such that |x - F_n| > 1

That means you have an infinite series with all terms positive and infinitely many terms greater than 1. So it diverges. f(x) is infinite for all x.

Did I misunderstand something?