Question

there a smallest real number a for which x26x is big-o of ax? explain your answer. (b) (2 points) is there a smallest integer number a for which x26x is big-o of ax? explain your answer.

Answer 1

No smallest real number exists for which x^2+6x is Big O of ax; a real number slightly larger than 1 would suffice. However, for integer values, the smallest integer a for which x^2+6x is Big O of ax is 2, because x^2 grows faster than ax whenever a<2.

Step-by-step explanation:

Big O Notation and Real Numbers

When considering whether x2+6x is Big O of ax, a clear definition of Big O notation is required. Big O notation is a way to describe the upper bound of a function in terms of another, often simpler, function. Specifically, f(x) is O(g(x)) if there exist positive constants C and k such that |f(x)| ≤ C|g(x)| for all x ≥ k. In this case, we need to find a real number a so that x2+6x ≤ C * ax for x sufficiently large. Conceptually, as x grows, x2 will eventually dominate 6x, meaning a has to be at least as big as 1 to account for the term x2. No smallest real number a exists because you can always choose another smaller number that would also work for a sufficiently large C.

Big O Notation and Integer Numbers

If we specifically look for the smallest integer number a for which x2+6x is Big O of ax, then the rules change slightly, since integers are not as flexible as real numbers. The smallest integer value of a that makes x2+6x ≤ C * ax for all sufficiently large x will be 2. This is because, simply put, the term x2 cannot be bounded by ax with a<1 since x2 grows faster than ax in that scenario. However, with a ≥ 2, it becomes possible to find a constant C that works.

Answer 2

No smallest real number a can make f(x) = x + 6x Big-O of g(x) = ax because Big-O defines an upper bound and any a > 7 would suffice, which means there is no single smallest value for a.

Step-by-step explanation:

The question asks whether there exists a smallest real number a for which the function f(x) = x + 6x is Big-O of g(x) = ax. For part (a), we can say that such a smallest real number does not exist because in Big-O notation, we are concerned with the upper bound of the growth rate of the function as x approaches infinity. As both terms in f(x) are linear, any a larger than 7 would suffice, making f(x) Big-O of g(x), and there is no single smallest value since we can always find a slightly smaller a that still satisfies the Big-O condition but is not the absolute smallest.