This is a Test of Mathematics Solution Subjective 150 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.
Also visit: I.S.I. & C.M.I. Entrance Course of Cheenta
Find the maximum among .
Consider the function . We employ standard techniques to compute the maxima.
Take logarithm on both sides we have . Next find out the derivative:
Since is always positive for positive x and so is sign of the derivative depends only on (1-logx). Hence the derivative is 0 at x = e (2.71 approximately), positive before that and negative after that. Hence the function has a maxima at x = e.
We check the values at x=2 and x=3 and easy computations show that . Hence is the largest value.
One may ask for a non calculus proof of this problem. The basic idea is to understand that the inequality
It is easy to show that the quantity lies within 2 and 3 for all values of n. Hence the inequality is true for n > 3. The result follows.