Use the Master theorem to find Θ bounds for each of the recurrence relations below. If the Master theorem does not apply, say so and explain why. You need to justify your answers to get full credit for this question. If you are applying Case 1 or Case 3 of the Master theorem, you need to show that one of the functions grows polynomially faster than the other and specify the ϵ for which it is true. If you are applying Case 3 of the Master theorem, make sure to check the regularity condition.
(a) T(0) = T(1) = C1
T(n) = 8T(n/2) + n³ + n*log n + 2ˡᵒᵍⁿ
(b) T(0) = T(1) = C1
T(n) = 2T(n/4) + n³/²
(c) T(0) = T(1) = C1
T(n) = 9T(n/3) + n³ + n² log n

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Write the command that can be used to answer the following questions. (hint: try each out on the system to check your results.) a. find all files on the system that have the word test" as part of their filename. b. search the path variable for the pathname to the awk command. c. find all files in the /usr directory and subdirectories that are larger than 50 kilobytes in size. d. find all files in the /usr directory and subdirectories that are less than 70 kilobytes in size. e. find all files in the / directory and subdirectories that are symbolic links. f. find all files in the /var directory and subdirectories that were accessed less than 60 minutes ago. g. find all files in the /var directory and subdirectories that were accessed less than six days ago. h. find all files in the /home directory and subdirectories that are empty. i. find all files in the /etc directory and subdirectories that are owned by the group bin."