Asymptotic notation

Question

Consider $f(n), g(n)$ and $h(n)$ be function defined as follows:
\[
\begin{array}{l}
f(n)=\Omega\left(n^{3}\right) \\
g(n)=O\left(n^{2}\right) \\
h(n)=\Theta\left(n^{2}\right)
\end{array}
\]

Which of the following represents correct asymptotic solution for $f(n)+[g(n) \times h(n)] ?$

• A. $\Omega\left(n^{4}\right)$
  
• B. $O\left(n^{4}\right)$
  
• C. $\Theta\left(n^{4}\right)$
  
• D. $O\left(n^{3}\right)$

Comments

Is it b? Bigoh(n⁴)

Basically we need to look for constant here ;

Let f=2n³

g=1/2n²

h=n²

Now eqn would be : 2n³+1/2n⁴ which is 1/2n⁴ thus it is O(n⁴)

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https://gateoverflow.in/172551/time-complexity

Answer 1

is option B) is the answer???

Comments

but, here the individual bounds are  ≤ n²  and  n²  respectively .So, for their product we can say it is n²

so why we are multilying here

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F(n)= n^5 also satisfy than how can we sa O(n^4)

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If F(n)=n^5 then answer will be be O(n^5)

but it is not there in the option......so you have to take the values according to the options.

Answer 2

The correct answer is B

Let assume f(n) = n ^ 4

                   g(n) = n ^ 2

                  f(n) = n ^ 2

after putting these values, you will get the answer.

Someone can say that it will fail for f(n) = n^5. But see the interesting answer it  will work for  f(n) = n ^ 3 and f(n) = n ^ 4 But if you see the other option they will fail more easily like option A for f(n) = n ^ 3 , g(n) = n , h(n) = n ^ 2. similarly you can find for the others. So I think the correct answer is B.