GATE CSE 2021 Set 1 Q13: Order 10^n, n^(log n), n^(sqrt n) by growth rate

Consider the following three functions.

$f_1 = 10^n \qquad f_2 = n^{\log n} \qquad f_3 = n^{\sqrt{n}}$

Which one of the following options arranges the functions in the increasing order of asymptotic growth rate?

Try it: multiple choice: exactly one correct option

Wrong answer costs 1/3 mark in GATE.

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Answer: D (official GATE 2021 final answer key)

Concept used: Compare growth rates by comparing logarithms.

First step: Take logs: $\log f_1 = n \log 10$ , $\log f_2 = (\log n)^2$ , $\log f_3 = \sqrt{n} \log n$ .

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Source: official GATE 2021 Set 1 CS question paper and final answer key, organised by IIT Bombay. Question paper · Answer key