This is a very beautiful sample problem from ISI MStat PSB 2007 Problem 3 based on use of L'hospital Rule . Let's give it a try !!
Let f be a function such that 
and f has derivatives of all order. Show that 
where 
is the second derivative of f at 0.
Differentiability
Continuity
L'hospital rule
Let L= 
it's a 
form as f(0)=0 .
So , here we can use L'hospital rule as f is differentiable .
We get L= 
= 
, taking -h=k .
= 
= . Hence done!
Let ![f:[0,1] \rightarrow[0,1]](https://www.cheenta.com/wp-content/ql-cache/quicklatex.com-871df19f08ccf0db083c99c935f6793f_l3.png "Rendered by QuickLaTeX.com")
be a continuous function such 
and assume that there exists a positive integer m such that 
for all ![x \in[0,1] .](https://www.cheenta.com/wp-content/ql-cache/quicklatex.com-34a5a961300479b1740d6dad2e0f684f_l3.png "Rendered by QuickLaTeX.com")
Prove that 
for all ![x \in[0,1]](https://www.cheenta.com/wp-content/ql-cache/quicklatex.com-c2a35f1975063fcb80056e3b34ed0065_l3.png "Rendered by QuickLaTeX.com")
This is a very beautiful sample problem from ISI MStat PSB 2007 Problem 3 based on use of L'hospital Rule . Let's give it a try !!
Let f be a function such that and f has derivatives of all order. Show that
where is the second derivative of f at 0.
Differentiability
Continuity
L'hospital rule
Let L= it's a form as f(0)=0 .
So , here we can use L'hospital rule as f is differentiable .
We get L=
= , taking -h=k .
= = . Hence done!
Let be a continuous function such and assume that there exists a positive integer m such that for all Prove that for all
