avissidep

2021-08-14

Use the definition to show that each function is $O\left({n}^{2}\right).$ O is big O notation

$F\left(n\right)=\mathrm{ln}\left(n\right)+5{n}^{2}$

hesgidiauE

Skilled2021-08-15Added 106 answers

Solution:
$f\left(n\right)=\mathrm{ln}\left(n\right)+5{n}^{2}$

$\left|f\left(n\right)\right|=|\mathrm{ln}\left(n\right)+5{n}^{2}|$

$\le \left|\mathrm{ln}\left(n\right)\right|+\left|5{n}^{2}\right|$

$\le \left|\frac{1}{2}\mathrm{ln}\left({n}^{2}\right)\right|+\left|5{n}^{2}\right|$

$\le \left|\frac{1}{2}{n}^{2}\right|+\left|5{n}^{2}\right|$

$\le \frac{11}{2}\left|{n}^{2}\right|$

Finally answer:$c=\frac{11}{2},f\left(n\right)isO\left({n}^{2}\right)$

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