7. Let f(n)>_f g(n) denote that the first derivative of f(n) is larger than that of g(n). (25) Let f(n)>_a g(n) denote that f(n) is asymptotically larger than g(n)s.

Let f(n)>_p g(n) denote that f(n) is polynomially larger than g(n). Note that exponentially larger also qualifies as polynomially larger.

Similarly, _f<_{, a}<_{, p}< are defined.

Fill in the blanks with one or more of (>_f, >_a, >_p, _f<, _a<, _p<, q) and briefly justify.

(i) n log n ___________ n

(ii) n log n ___________ n^1.1

(iii) 2ⁿ ________________ n²⁰

(iv) 2ⁿ __________________ n!

(v) n³ + 3n²+1000 ____________ n log n + n³/6