a$)3$ logn $+4$n log ntn is in O$($n logn$)$ b$)2$n $+14$n$2+4$n$3$ is in $1(21)$ c$)546+34$n $+2$n$2$ is $($n$?)$ a$)3$logn $+4$nlogn $+$n is in O$($nlogn$)$ Using T$($n$)$ is O$($f$($n$))$ if there exist two positive constants c and no such that T$($n$)<$ cf$($n$)$ for all n $>$ no: $3$logn $+4$nlogn$+$ns O$($nlogn$)=3$logn $+4$nlogn$+$nSc$*$ nlogn$,$ where n$>3=3$log$(3)+4(3)$log$(3)+3$ Sc$*3$log$(3)=10\mathrm{.}16$ c$*1\mathrm{.}43$ ~ $7\mathrm{.}1$ sc $=$c$=7\mathrm{.}1$ and no $=3$ b$)2$n $+14$n$2+4$n$3$ is in $2(21)$ Using T$($n$)$ is $2($g$($n$))$ if there exist two positive constants c and no such that T$($n$)>$ cg$($n$)$ for all n $>$ no: $2$n $+14$n$2+4$n$3>2(21)=2$n $+14$n$2+4$n$3>$ c$*2$n$,$ where n $28=28+14(8)2+4(8)32$ c$*28=20/2$ c $=1020=$c$=10$ and no $=8$ c$)546+34$n $+2$n$?$ is in O$($na$)$ Using f$($n$)=0($g$($n$))$ if there exist positive constants c$1,$ c$2$ and no such that ci g$($n$)=$ f$($n$)=$ C$2$ g$($n$)$ for all n $>$ no: ci g$($n$)<546+34$n$+2$n$2<$ c$2$ g$($n$)=$c$*$ n$<546+34$n $+2$n$?,$ where n $>1=$c$*(1)2=546+34(1)+2(1)2=$ci $<582=546+34$n$+2$n$?$ Sc$2*$ n$2,$ where n $21=546+34(1)+2(1)25$ c$2*(1)2=5825$ C$2=$ci $<582,$ c$2>582$ and no $=1$