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# ISI MStat PSB 2007 Problem 3 | Application of L'hospital Rule 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 !!

## Problem- ISI MStat PSB 2007 Problem 3

Let f be a function such that $f(0)=0$ and f has derivatives of all order. Show that $\lim _{h \to 0} \frac{f(h)+f(-h)}{h^{2}}=f''(0)$
where $f''(0)$ is the second derivative of f at 0.

### Prerequisites

Differentiability

Continuity

L'hospital rule

## Solution :

Let L= $\lim _{h \to 0} \frac{f(h)+f(-h)}{h^{2}}$ it's a $\frac{0}{0}$ form as f(0)=0 .

So , here we can use L'hospital rule as f is differentiable .

We get L= $\lim _{h \to 0} \frac{f'(h)-f'(-h)}{2h} = \lim _{h \to 0} \frac{(f'(h)-f'(0)) -(f'(-h)-f'(0))}{2h}$

= $\lim _{h \to 0} \frac{f'(h)-f'(0)}{2h} + \lim _{k \to 0} \frac{f'(k)-f'(0)}{2k}$ , taking -h=k .

= $\frac{f''(0)}{2} + \frac{f''(0)}{2}$ = $f''(0)$ . Hence done!

## Food For Thought

Let $f:[0,1] \rightarrow[0,1]$ be a continuous function such $f^{(n)} := f ( f ( \cdots ( f(n \text{ times} ))$ and assume that there exists a positive integer m such that $f^{(m)}(x)=x$ for all $x \in[0,1] .$ Prove that $f(x)=x$ for all $x \in[0,1]$

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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 !!

## Problem- ISI MStat PSB 2007 Problem 3

Let f be a function such that $f(0)=0$ and f has derivatives of all order. Show that $\lim _{h \to 0} \frac{f(h)+f(-h)}{h^{2}}=f''(0)$
where $f''(0)$ is the second derivative of f at 0.

### Prerequisites

Differentiability

Continuity

L'hospital rule

## Solution :

Let L= $\lim _{h \to 0} \frac{f(h)+f(-h)}{h^{2}}$ it's a $\frac{0}{0}$ form as f(0)=0 .

So , here we can use L'hospital rule as f is differentiable .

We get L= $\lim _{h \to 0} \frac{f'(h)-f'(-h)}{2h} = \lim _{h \to 0} \frac{(f'(h)-f'(0)) -(f'(-h)-f'(0))}{2h}$

= $\lim _{h \to 0} \frac{f'(h)-f'(0)}{2h} + \lim _{k \to 0} \frac{f'(k)-f'(0)}{2k}$ , taking -h=k .

= $\frac{f''(0)}{2} + \frac{f''(0)}{2}$ = $f''(0)$ . Hence done!

## Food For Thought

Let $f:[0,1] \rightarrow[0,1]$ be a continuous function such $f^{(n)} := f ( f ( \cdots ( f(n \text{ times} ))$ and assume that there exists a positive integer m such that $f^{(m)}(x)=x$ for all $x \in[0,1] .$ Prove that $f(x)=x$ for all $x \in[0,1]$

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