able 2.1 contains values of several functions that often arise in the analysisof algorithms. These values certainly suggest that the functionslog n, n, n log 2 n, n2, n3, 2n , n!are listed in increasing order of their order of growth. Do these valuesprove this fact with mathematical certainty?b. Prove that the functions are indeed listed in increasing order of their orderof growth.

The text you provided seems to be incomplete, and it's hard to provide a comprehensive answer without the full context. However, I can give you a general idea of how to approach this problem.

a. The values in the table do not prove with mathematical certainty that the functions are listed in increasing order of their order of growth. They only suggest this might be the case. To prove it with mathematical certainty, you would need to use mathematical analysis and proofs.

b. To prove that the functions are indeed listed in increasing order of their order of growth, you would need to use the definitions and properties of Big O notation, which is used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, in this case as n tends towards infinity.

Here's a general idea of how you might approach this:

1. Start with the first two functions, log n and n. It's a well-known fact in computer science that n grows faster than log n, so n is indeed bigger than log n in terms of order of growth.

2. Next, compare n and n log 2 n. The function n log 2 n grows faster than n, because it's n times a growing function (log 2 n).

3. Continue this process with the rest of the functions. For example, n^2 grows faster than n log 2 n, n^3 grows faster than n^2, 2^n grows faster than n^3, and n! grows faster than 2^n.

Remember, these are just general ideas. To fully prove these facts, you would need to use mathematical proofs involving the definitions and properties of Big O notation.