#made easy

Question

Big oh estimate for

f(x)=(x+1)log($x^2 +1$)+3$x^2$ is given as

1.O(xlogx)

2.O($x^2$)

3.O($x^3$)

4O($x^2$logx)

Answer 1

Here, first term in the right hand side, $(x+1)\log (x^2 + 1) = O(x \log x)$
now, second term is $O(x^2)$
$x^{2} > x \log x$ for $\forall$ x > $n_0$ where $n_0$ is positive constant
so,
f(x) = $O(x^2)$

Comments

$x^2 > x \log x, \forall x > n_0$ is the condition. Not some $x$.

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Yes sir corrected.