ISI MStat PSB 2007 Problem 4 | Application of Newton Leibniz theorem
Join Trial or Access Free ResourcesJoin Trial or Access Free ResourcesThis is a very beautiful sample problem from ISI MStat PSB 2007 Problem 4 based on use of Newton Leibniz theorem . Let's give it a try !!
Let \( f: \mathbb{R} \rightarrow \mathbb{R}\) be a bounded continuous function. Define \( g:[0, \infty) \rightarrow \mathbb{R} \) by,
\( g(x)=\int_{-x}^{x}(2 x t+1) f(t) dt \)
Show that g is differentiable on \( (0, \infty) \) and find the derivative of g.
Riemann integrability
Continuity
Newton Leibniz theorem
As \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a bounded continuous function hence the function
\( |\Phi(t)|=|(2xt+1)f(t)|=|2xt+1||f(t)|<(|2xt|+1)M<(2|x|^2+1)M \) , which is finite for a particular x so it's a riemann integrable function on t.
Now, by fundamental theorem we have g(x)=F(x)-F(-x) , where F is antiderivative of \( \Phi(t) \) .
Hence from above we can say that g(x) is differentiable function over x .
Now by Leibniz integral rule we have \( g'(x)=(2x^2+1)f(x)+f(-x)(1-2x^2) + \int_{-x}^{x} (2t)f(t) dt \).
Let \( f: \mathbb{R} \rightarrow \mathbb{R}\) be a continuous function. Now, we define \(g(x)\) such that \( g(x)=f(x) \int_{0}^{x} f(t) d t \)
Prove that if g is a non increasing function, then f is identically equal to 0.
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.