Big – O estimate for \(f(x) = (x + 1) \ log(x^2 + 1) + 3x^2\) is given as
2013
Big – O estimate for
\(f(x) = (x + 1) \ log(x^2 + 1) + 3x^2\)
is given as
- A.
\(O(x \ log \ x)\) - B.
\(O(x^2)\) - C.
\(O(x^3)\) - D.
\(O(x^2 \ log \ x)\)
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Correct answer: B
Answer: f(x) is O(x^2).
Simplify the logarithmic part: for large x, log(x^2 + 1) = 2·log x + o(1), so (x + 1)·log(x^2 + 1) = 2x·log x + lower-order terms = O(x·log x).
Compare with the polynomial term: 3x^2 is Θ(x^2) and grows faster than x·log x, so 3x^2 dominates the sum for large x.
Conclusion: f(x) = 3x^2 + O(x·log x) = Θ(x^2), therefore f(x) is O(x^2).