Try this problem of TIFR GS-2010 from Real analysis, Differentiantiation and Maxima and Minima.

REAL ANALYSIS | TIFR 201O| PART A | PROBLEM 5


The maximum value of $f(x)=x^n(1-x)^n$ for natural number $n\geq 1$ and $0\leq x\leq1$

  • $\frac{1}{2^n}$
  • $\frac{1}{3^n}$
  • $\frac{1}{5^n}$
  • $\frac{1}{4^n}$

Key Concepts


REAL ANALYSIS

MAXIMA AND MINIMA

DIFFERENTIATION

Check the Answer


But try the problem first…

Answer:$\frac{1}{4^n}$

Source
Suggested Reading

TIFR 2010|PART A |PROBLEM 1

AN INTRODUCTION TO ANALYSIS DIFFERENTIAL CALCULUS PART-I RK GHOSH, KC MAITY

Try with Hints


First hint

Here first differentiate $f(x)$

Second Hint

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 $n$

Final Step

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 $\frac{1}{4^n}$

Subscribe to Cheenta at Youtube