Prove the above questions BigO and BigOmega Notation identif

Prove the above questions (Big-O and Big-Omega Notation.).... ...identifiy constants or tell me why not possible and exaplain why proof fails. please prove them with inequlity and explain percisly.... thanks

Solution

Answer:

5n^2/8 belongs to omega (n^3)

we already know the definition of the omega as : -

f(n) >= c*g(n) , where c is constant and is greater than 0.

here , f(n) = 5n^2/8 , g(n) = n^3

5n^2/8 > = n^3

=> 5n^2/ 8 - n^3 = 0

=> n^2(5/8-n) = 0

=> 5/8 - n = 0

=> n = 5/8

Hence 5n^2/8 is omega(n^3) at n = 5/8