Arrange the following functions in increasing asymptotic order:
A. n^1/3
B. eⁿ
C. n^7/4
D. n log⁹n
E. 1.0000001ⁿ
a)
A, D, C, E, B
b)
D, A, C, E, B
c)
A, C, D, E, B
d)
A, C, D, B, E
Correct answer is option 'A'. Can you explain this answer?

Question

Arrange the following functions in increasing asymptotic order:
A. n^1/3
B. eⁿ
C. n^7/4
D. n log⁹n
E. 1.0000001ⁿ

a)
A, D, C, E, B
b)
D, A, C, E, B
c)
A, C, D, E, B
d)
A, C, D, B, E

Correct answer is option 'A'. Can you explain this answer?

Answer

Explanation:

To arrange the given functions in increasing asymptotic order, we need to compare their growth rates as n approaches infinity.

1. Constant function: 1.0000001n
This is a constant function, as it does not depend on the value of n. Therefore, its growth rate is O(1), which is the slowest possible growth rate.

2. Cube root function: n1/3
As n approaches infinity, the cube root of n grows slower than any positive power of n, but faster than any logarithmic function. Therefore, its growth rate is O(n1/3).

3. Logarithmic function: n log9n
As n approaches infinity, the logarithmic function grows slower than any positive power of n, but faster than any root function. However, the base of the logarithm does not affect its growth rate, so we can ignore the base 9. Therefore, its growth rate is O(n log n).

4. Power function: n7/4
As n approaches infinity, any positive power of n grows faster than any root or logarithmic function. Therefore, its growth rate is O(n7/4).

5. Exponential function: en
As n approaches infinity, the exponential function grows faster than any positive power of n, root function, or logarithmic function. Therefore, its growth rate is O(en).

Arranged order from slowest to fastest growth rate:
A. n1/3
D. n log9n
C. n7/4
E. en
B. 1.0000001n

Therefore, the correct order is option 'A': A, D, C, E, B.

Answer

A

Answer

Ans c is the correct one

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