Big $O(25$ points$)$ This problem is on determining whether one function is Big$-$O of the other.

Provide a proof for your answers. If you are showing that if finO$(g),$ then you must show

that there is a constant $c1$ such that for every $n1,f(n)cg(n).$ Your proof must state

the value of the constant $c,$ and, if your inequality only holds for values of $n$ that are above

some threshold $N,$ then you must provide the value of $N$ as well.

If you are showing that $f!$inO$(g),$ then you must show that for every constant $c1$ and

every threshold $N>0,$ there is always an $n>N$ for which $f(n)cg(n).($Alternatively$,$ you

may provide a proof by contradiction.$)$

In your proofs, you may use the fact that for every $k>0$ and $a>0,n^k$inO$(n^{k+a}),$ and

$lo{g}^k$ninO$(n^a).$

$($a$)$ Is $9n^5+3n^3+7$inO$(n^5)?$

$($b$)$ Is $\sqrt[\phantom{2}]{n}-1$inO$(lo{g}_3(n))?$

$($c$)$ Is $2^{2^{n+1}}$inO$(2^{2^n})?$

$($d$)$ Prove or disprove: If $f(n)inO(g(n))$ then $logf(n)inO(logg(n)).$