You have been invited to the panel of judges for the most efficient business app category. Show how you would rank each of the following four (4) business apps, from 1 (most efficient) to 4 (least efficient) using Big O notation. Note that \( c \) in each of the algorithms below represents the number of customers that the business app will have to process. Show all your workings and ranking scale.

App a: [tex] f(c) = 84 + 59c \log(c) + 150c + 590c \log(c) + 203c + 5900c \log(c) [/tex]

App b: [tex] f(c) = 412c + (111 \times 10c) + 2048 [/tex]

App c: [tex] f(c) = 9c^5 + 7c^2 + 351c + c + \frac{1}{2} c^2 + \frac{1}{4} c^2 [/tex]

App d: [tex] f(c) = \frac{1}{2} c^2 + \frac{1}{4} c^2 + \frac{1}{8} c^2 + \frac{1}{16} c^2 + \frac{1}{32} c^2 + 7c^2 [/tex]

Analyze and rank these apps based on their efficiency using Big O notation.

The task at hand is to rank four business apps based on their efficiency using Big O notation. Big O notation is a mathematical notation that helps classify functions based on their rate of growth.

Let's evaluate each app and rank them accordingly:
App a) f(c) = 84 + 59c(log(c)) + 150c + 590c(log(c)) + 203c + 5900c(log(c))

To determine its efficiency, we need to focus on the term that grows the fastest as the value of 'c' increases.

In this case, we have terms like 59c(log(c)), 590c(log(c)), and 5900c(log(c)).

The logarithmic terms will grow at a slower rate compared to the other terms.

Therefore, the highest power term that will dominate the growth rate is 5900c(log(c)).

App b) f(c) = 412c + (111 * 10c) + 2048

In this app, the highest power term is 111 * 10c.

As the value of 'c' increases, this term will dominate the growth rate.
App c) f(c) = 9(c^5) + 7(c^2) + 351c + c + 1/2(c^2) + 1/4(c^2)

Here, the highest power term is 9(c^5).

As 'c' increases, the growth rate of this term will dominate.
App d) f(c) = 1/2(c^2) + 1/4(c^2) + 1/8(c^2) + 1/16(c^2) + 1/32(c^2) + 7(c^2)

In this app, the highest power term is 7(c^2). As 'c' increases, the growth rate of this term will dominate.

Ranking the apps based on their growth rate, we have:

1. App c: The highest power term is 9(c^5), indicating the slowest growth rate.
2. App d: The highest power term is 7(c^2), which grows faster than App c.
3. App b: The highest power term is 111 * 10c, indicating a faster growth rate compared to App d.
4. App a: The highest power term is 5900c(log(c)), suggesting the fastest growth rate among the four apps.

Therefore, the ranking from most efficient to least efficient is:
1. App c
2. App d
3. App b
4. App a

By using Big O notation, we can compare the growth rates of different algorithms and determine their relative efficiencies.

Know more about notation here:

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