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 \(f(0)=0\) 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.
Differentiability
Continuity
L'hospital rule
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!
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 \( x \in[0,1] \)
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]
In 2025, 8 students from Cheenta Academy cracked the prestigious Regional Math Olympiad. In this post, we will share some of their success stories and learning strategies. The Regional Mathematics Olympiad (RMO) and the Indian National Mathematics Olympiad (INMO) are two most important mathematics contests in India.These two contests are for the students who are […]
Cheenta Academy proudly celebrates the success of 27 current and former students who qualified for the Indian Olympiad Qualifier in Mathematics (IOQM) 2025, advancing to the next stage — RMO. This accomplishment highlights their perseverance and Cheenta’s ongoing mission to nurture mathematical excellence and research-oriented learning.
