a) Prove that

f(n)=3n^(2)+2n+1

is in

O(n^(2))

. Show the constants

c

and

n_(0)

that satisfy the\ definition of Big 0 .\ b) Prove that

f(n)=n^(3)+4n^(2)+2n

is in

\\\\Theta (n^(3))

. Show the constants

c1,c2

, and

n_(0)

that satisfy\ the definition of Theta notation.\ c) Prove that

f(n)=2n^(2)+7n

is in

\\\\Omega (n^(2))

. Show the constants

c

and

n0

that satisfy the\ definition of Omega notation.\ d) Given two functions

g(n)=n^(2)

and

h(n)=n^(3)

, determine whether ). Provide a\ proof for your answer.\ e) For the function

f(n)=5n^(2)+3nlogn+10

, determine the tightest possible bound using\ Big O, Omega, and Theta notations. Explain your reasoning.\ Remember, proofs for asymptotic notations often involve finding appropriate constants (c\ and

n_(0)

) that satisfy the definitions of Big

O

, Theta, and Omega. Provide clear explanations\ and reasoning in your answers.

a) Prove that f(n)=3n2+2n+1 is in O(n2). Show the constants c and n0 that satisfy the definition of Big O. b) Prove that f(n)=n3+4n2+2n is in (n3). Show the constants c1,c2, and n0 that satisfy the definition of Theta notation. c) Prove that f(n)=2n2+7n is in (n2). Show the constants c and n0 that satisfy the definition of Omega notation. d) Given two functions g(n)=n2 and h(n)=n3, determine whether g(n)=O(h(n) ). Provide a proof for your answer. e) For the function f(n)=5n2+3nlogn+10, determine the tightest possible bound using Big O, Omega, and Theta notations. Explain your reasoning. Remember, proofs for asymptotic notations often involve finding appropriate constants (c and n0 ) that satisfy the definitions of Big O, Theta, and Omega. Provide clear explanations and reasoning in your answers