GATE CS 2022 Set 1 — Question 11
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Question
Which one of the following statements is TRUE for all positive functions ?
Correct answer
(A) f(n²) = θ(f(n)²), when f(n) is a polynomial
Solution
Let's analyze the options:(A) If is a polynomial, let for some constants .
Then .
And .
Since , . This statement is TRUE.(C) If is an exponential function, let .
Then .
And .
Since grows much faster than , . This statement is FALSE.(B) and (D) are not true for all positive functions (counterexamples can be found using the functions above).
Then .
And .
Since , . This statement is TRUE.(C) If is an exponential function, let .
Then .
And .
Since grows much faster than , . This statement is FALSE.(B) and (D) are not true for all positive functions (counterexamples can be found using the functions above).
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