• Support PF! Buy your school textbooks, materials and every day products Here!

L'hopitals rule, f''(a)

  • Thread starter Kate2010
  • Start date
  • #1
146
0

Homework Statement



I have to show that limh->0 [(f(a+h)-2f(a)+f(a-h))/h2] = f''(a) where f:R->R is differentiable, a is a real number and f''(a) exists.

Homework Equations





The Attempt at a Solution



I have applied l'hopitals rule as the question advises and have got to f''(a). However, my problem lies in checking the conditions for l'hopitals rule each time. I'm ok with showing the f(x) (numerator) and g(x) are zero when x=0 for each case. However, I am unsure which region I can take for differentiable and continuous. I thought maybe differentiable on (-2a,2a) \ {0} and continuous on [-2a,2a]. I have used these regions each time. But I am also unsure if I can claim that as f''(a) exists then f' is differentiable in the region (-2a,2a)/{0} and continuous in [-2a,2a]. If f''(a) exists, f' is not necessarily differentiable in this region? Or is it? I wasn't sure what else I could do.
 

Answers and Replies

  • #2
146
0
Or would it be better to say continuous in the region [a-2h,a+2h], differentiable in the region (a-2h,a+2h)\{0}. Is this allowed even though h is tending to 0?
 

Related Threads on L'hopitals rule, f''(a)

  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
2
Views
827
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
12
Views
4K
  • Last Post
Replies
8
Views
5K
  • Last Post
Replies
13
Views
2K
  • Last Post
Replies
1
Views
783
  • Last Post
Replies
12
Views
1K
  • Last Post
Replies
4
Views
1K
Top