$$\text { Show that the function } f ( x ) = ( x + 2 ) \log \left( x ^ { 2 } + 1 \right) + \log \left( x ^ { 3 } + 1 \right) \text { is } O ( x \log x ) \text {. }$$

Name: \[\text { Show that the function } f ( x ) = ( x + 2 ) \log \left( x ^ { 2 } + 1 \right) + \log \left( x ^ { 3 } + 1 \right) \text { is } O ( x \log x ) \text {. }\]
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Question

Quizplus · Accepted Answer

The Answer is We have

x + 2

is

O ( x )

since

x + 2 \leq 2 x

for all

x \geq 2

;

\log \left( x ^ { 2 } + 1 \right)

is

O ( \log x )

since

\log \left( x ^ { 2 } + 1 \right) \leq \log \left( 2 x ^ { 2 } \right) =

\log 2 + 2 \log x \leq 3 \log x

whenever

x \geq 2

; and similarly

\log \left( x ^ { 3 } + 1 \right)

is

O ( \log x )

. It follows that

( x + 2 ) \log \left( x ^ { 2 } + 1 \right)

is

O ( x \log x )

and consequently

f ( x )

is

O ( x \log x )

.