asymptotic tight bound for quadratic functions

Question

In CLRS (Introduction to Algorithms by Cormen, Leiserson, Rivest, and Stein), for a function

f(n) = an² + bn

Answer 1

well its easy 1.c1<=a + b/n + c/n^2 Now here a is >0 while b,c are either positive or negative Now we must choose value of n such that b/n + c/n^2 never exceeds a in RHS of above eqution cause else it will become negative and so will c1. but by definition c1 is a positive constant

So we want to make sure a>b/n+c/n^2

if we choose n=2*max(|b|/a, sqrt(|c|/a) ) we will get b/n + c/n^2 as an expression whose value is less than a/2+a/4.

thus a+b/n+c/n^2 will have maximum value as a+a/2+a/4 and minimum value as a-(a/2+a/4) which is nothing but values of c2 and c1.

c1=a-a/2-a/4= a/4 c2=a+a/2+a/4= 7a/4

This concept can be extended to any values for any polynomial..

cheers!!!