- $f_{1}(n) = n^{100}$
- $f_{2}(n) = (1.2)^{n}$
- $f_{3}(n) = (2)^{n/2} = (1.414)^{n}$
- $f_{4}(n) = (3)^{n/3} = (1.442)^{n}$
We know that $(1.442)^{n} > (1.414)^{n}$ and $(1.442)^{n} > (1.2)^{n}$
$\Rightarrow$$f_{4}(n)$ is greater than both $f_{2}(n)$ and $f_{3}(n)$
Now we have to check between $f_{4}(n)$ and $f_{1}(n)$
But $f_1(n)$ is a polynomial function and $f_4(n)$ is a exponential function. Any exponential function will have larger value than any polynomial function for sufficiently large value of input. So, $f_4(n)$ has the largest value here.
Option A. $f_{4}(n)$ is the correct answer.