I've gotten through the entirety of this assignment and about 6 other proofs without any problem, but I'm just struggling with these two. Both of these are in a section that heavily covers the Mean Value Theorem. I'm not looking for answers directly, but maybe pointers could be nice. All of my friends are artsy and avoid math like the plague.
(Edit here.. I had the problem written down slightly wrong.)
I've gotten as far as..If f is a differentiable, odd function show that ∀b>0, ∃c∈(-b,b) st f'(c) = f(b)/b
f(-b) - f(b) = f'(c)(-b - b)
f'(c) == (f(-b) - f(b))/(-b - b)
I've gotten the second. So others may take a look at this later for any reason, I'll leave my work here:
Prove |sin a - sin b| ≤ |a - b|
sin a - sin b = d/dc(sin c)(a - b) by Mean Value Theorem
sin a - sin b = cos c (a - b)
|sin a - sin b| = |cos c||a - b|
|sin a - sin b| ≤ |a - b| since |cos c| >= 0
If I need to post these both on different threads, I'll do so. I was a bit unclear about the section in the rules saying new questions need new threads, and if that means that two questions had to have their own.
Ahhh, you're right. I'm not sure where that came from, but it's f(b)/b, not f'(b).
Thanks. I'll update the question and take a closer look.
Edit: Your tip also definitely helped with the second one.
I'm still not much further with the first, since I know I have to use the fact that it's an odd function, but am not quite sure where.