Discussion
Big - O estimate for \(f(x) = (x + 1)\log(x^{2} + 1) + 3x^{2}\) is given as
Answer: B
The estimate is O(x^2). For large x, log(x^2 + 1) is about 2 log x, so the first term (x + 1) log(x^2 + 1) grows like x log x, which is smaller than x^2. The second term, 3x^2, grows as x^2.
\[ f(x) = O(x \log x) + O(x^2) = O(x^2) \]The dominant term decides, so O(x log x) is too small and O(x^3) and O(x^2 log x) are valid upper bounds but not the tightest. The standard estimate is O(x^2).
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