Try this problem of TIFR GS-2010 from Real analysis, Differentiantiation and Maxima and Minima.
The maximum value of $f(x)=x^n(1-x)^n$ for natural number $n\geq 1$ and $0\leq x\leq1$
REAL ANALYSIS
MAXIMA AND MINIMA
DIFFERENTIATION
Answer:$\frac{1}{4^n}$
TIFR 2010|PART A |PROBLEM 1
AN INTRODUCTION TO ANALYSIS DIFFERENTIAL CALCULUS PART-I RK GHOSH, KC MAITY
Here first differentiate $f(x)$
Then equate the terms of $f'(x)$ containing $x$ to $0$ and find all possible values of $x$,since your answer is in terms of $n$ no need to perform any kind of operations on $n$
Now equating $x$ we get $x=0,\frac{1}{2},1$
Now put each of these values of $x$ in $f(x)$ and see for which value of $x$ you get the maximum value of $f(x)$
you will get the maximum value of $f(x)$ for $x=\frac{1}{2}$ that is $\frac{1}{4^n}$
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.