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Exercise 9.1 Quadratic Upper Bound:
Show that Tn = 3n^2 + 4n + 100 is O(n) by showing that T(n) ≤ 5n for all n > 10. What's f(n), c, and N for this problem?
Complete the proof by filling in the blanks:
3n^2 ≤ n^2 for n > 0
n^2 ≤ nforn > (Use smallest integer for which this is true)
nforn > (Use smallest integer for which this is true)
So, Tn = 3n^2 + 4n + 100 ≤ 3n^2 + n^2 + n^2 = 5n^2 for all n > 10.
Since T(n) = 3n^2 + 4n + 100 and n^2 ≤ nforn > 0 (and in particular for all n > 10), it follows from the definition of absolute value that T(n) = |Tn and n = nl, so by substitution in the above inequality, we get: |Tn| ≤ 5n for all n > 10.
Thus, we proved that T(n) is O(f(n)) from the above inequality where f(n) is n, c, and N.
