Standard Problem (that I can't do)

Let f(x) be a differentiable function from and to the reals. Given that f'(x) > 0 for all x, prove that f(x) is (strictly) increasing.

I can only get the function to be increasing locally around every point.

Reply 1

generalebriety

Post your proof so far?

Reply 2

DFranklin

Seeing as (unless I've got the poster completely wrong) this is something you did ages ago, as opposed to something you're currently studying (and so we might as well cut to the chase):

Spoiler

Reply 3

Drederick Tatum

It's gonna be annoying to write the symbols, but if f'(x) > 0 for all x then we can find a d > 0 such that for all h in (-d,d) we have -f'(x) < (f(x+h)-f(x))/h - f'(x) < f'(x). In particular, for all h in (0,d) we have f(x+h)>f(x).

So we have the statement "For all x there is a d such that for all h in (0,d) we have f(x+h)>f(x)."

Reply 4

Drederick Tatum

Thanks DFranklin. This has just come up in some stuff on interest rates in my actuarial work.

Reply 5

generalebriety

DFranklin

Seeing as (unless I've got the poster completely wrong) this is something you did ages ago

Yeah, I was wondering about that...

Reply 6

DFranklin

generalebriety

Yeah, I was wondering about that...

Sad reflection on how long people remember stuff after they learn it.

(Of course, it's 20 years since I did it, and I still remember it. Principles of Dynamics, on the other hand, I remember absolutely nothing about, which is a pity, as it would be useful knowledge for me these days).

[Going back to the question: it's trivial if you know which theorem to use, and pretty much impossible if you don't, which is why I figured there was no easy way of leading the poster through it].

Reply 7

Drederick Tatum

I'm trying to keep up with this stuff, but you lose a lot of the sharpness. It's a bit annoying to think I was probably at my best in this sense when I was doing STEP. I think that if I looked through the definitions and statements of theorems for an undergraduate course I would feel confident at proving everything in them. But I would be useless at tripos questions now.

I don't know how you manage to remember these sorts of things after 20 years.

Reply 8

DFranklin

Drederick Tatum

I don't know how you manage to remember these sorts of things after 20 years.

Partly it's that I knew Analysis really well, and partly it's that I kept using those results all the way up through Part III (and beyond - I got a job at DAMTP after part III and supervised analysis for a bit at the same time).

I know what you mean about the sharpness though.