Twice differentiability

1. Dec 6, 2008

tomboi03

a. Suppose f is twice differentiable on (0, infinity). Suppose that |f(x)|< or equal A0 for all x>0 and that the second derivative satisfies |f''(x)|< or equal A2 for all x>0.
Prove that for all x>0 and all h>0
|f'(x)| < or equal 2A0/h + A2h/2
This is sometimes called Landau's inequality.

b. Use part a to show that for all x>0
|f'(x)| < or equal 2 sqrt(A0A2)

Should I just try to do this problem backwards by trying to find what f(x) by using the f'(x)?

Thanks

2. Dec 6, 2008

HallsofIvy

Looks to me like the mean value theorem should work. If f is twice differentiable on the positive numbers, then you can apply the mean value theorem to f' on any interval of positive numbers: (f'(x)- f'(x0))/(x- x0)= f"(c) where c is between x0 and x. Taking x- x0= h, that is f'(x)- f'(x0)= f"(c)h.
Also use f(x)- f(x0)= f'(d)h, the mean value theorem applied to f.

3. Dec 10, 2008

tomboi03

I'm not sure how to do this, can you elaborate how to do this esp with the mean value theorem?

Thank you