Question

For the problem below it deals with finding the Upper Bound of equation T(n), what do these 3 lines mean? And how does this problem show us the Upper Bound?

T(n) = O(n²)

T(n) is O(n²)

T(n)

€O(n)

The actual problem:

Answer 1

By definition of big O complexity :

f(n) = O(g(n)) if there exists a positive integer n₀ and a positive constant c, such that f(n)≤c.g(n) ∀ n≥n₀

Here , f(n) = p*n² + q*n + r and g(n) = n²

As stated by you we can easily show that

f(n) <= c*g(n) where c = (p + q + r )

The three lines that you've mentioned actually refers to same thing that is the upper bound of T(n) is n²   

Talking about the question that what is the meaning of upper bound.

If we plot the graph of (p + q + r) * n² vs p*n² + q*n + r , the latter will lie below the former for all values of n.In other words,   (p + q + r) * n² >= p*n² + q*n + r   ∀ n

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Please comment in case you have any doubt or anything is left unclear in my answer.