$3-1$: $,$ Let $p(n)={\displaystyle \underset{i=0}{\overset{d}{}}}{a}_i{n}^i$

Where $a_d>0,$ be a degree$-d$ polynomial in $n,$ and let $k$ be a constant. Use the definitions of the asymptotic notations to prove the following properties.

a$)$ If $kd$ then $p(n)=O(n^k).$

b$)$ If $kd$ then $p(n)=(n^k).$

c$)$ If $k=d$ then $p(n)=(n^k).$

$3-3$: Rank the following functions by order of growth: $($either decreasing or increasing$).$

$(\frac{3}{2}{)}^n,2^{2n},n^3,2^{lgn},(lg(n))!$

Note: $lgn$ is $lo{g}_{2}n.$

$3-4$: Let $f(n)$ and $g(n)$ be asymptotically positive functions. Prove or disprove the following:

a$)f(n)=O(g(n))$ implies $g(n)=O(f(n)).$

b$)f(n)+g(n)=(min(f(n),g(n))).$

c$)f(n)=O(g(n))$ implies $2^{f(n)}=O(2^{g(n)}).$