$6.$ Big$-$O$-$ver it $[12$ points$]$

Give the tightest big$-$O estimate in simplest form for each of these functions and algorithms.

Show your steps to arrive at the final expression, and explain your answer.

$($a$)$ f $($n$)=($n$2+203$n$)($n$!+$ nn$)+($n$2+$ n$!)(203$n $+$ nn$)$

$($b$)$ g$($n$)=($log n $+$n$)(2$n $+3$n$)$

$4$

$($c$)$ function sumOddIndices$($a$1,...,$ aN in R$)$

i $1$

s $0$

while i $$ N do

s $$ s $+$ ai

i $$ i $+2$

end while

return s

end function

$($d$)$ function findMinPowerAboveN$($N in Z$+)$

i $1$

while i $$ N do

i $2$i

end while

return i

end function

$($e$)$ function findMaxDifference$($a$1,...,$ aN in R$)$

d $0$

for i :$=1$ to N do

for j :$=1$ to N do

if $|$ai $$ aj $|>$ d then

d $|$ai $$ aj $|$

end if

end for

end for

return d

end function