Problem $5(4+4+4=12)$ Growth OF FUNCTIONS

For each of these parts, indicate whether $f=O(g),f=(g),$ or both $($i$.$e$.,f=(g)).$ In each case, give a

justification for your answer. Your justification can be in the form of a proof from first principles or a proof

using limits$,$ and can use any of the facts presented in the lecture or the text. $($Hint: It may help to plot the

functions and obtain an estimate of their relative growth rates. In some cases, it may also help to express

each function as a power of $2$ and then compare.$)$

$($a$)f(n)=n{2}^n$;$g(n)=3^n.$

$($b$)f(n)=nlogn$;$g(n)=n^{\frac{4}{3}}.$

$($c$)f(n)=10n+(logn{)}^2$;$g(n)=100n-8\sqrt[\phantom{2}]{n}.$