Question

Use the basic definition (Ad-hoc Calculations) to show that f(x)=3x3−5x2+4x−85x5+3x3−x2−7​∈O(x2) for x∈R+.

Prove that f(x)∈Ω(x2) for x∈R+.

Answer 1

To prove that f(x) is in O(x^2) and Ω(x^2), we can divide the numerator and denominator by x^2 to simplify the expression. By choosing appropriate constants and showing that the absolute value of the expression is less than or equal to c|x^2| and greater than or equal to c|x^2|, we can demonstrate that f(x) belongs to both O(x^2) and Ω(x^2) for x ∈ R+.

Step-by-step explanation:

To prove that f(x) = 3x^3 - 5x^2 + 4x - 8 / (5x^5 + 3x^3 - x^2 - 7) is in O(x^2) for x ∈ R+, we need to show that there exist positive constants c and k such that for all x larger than some value x0, the absolute value of f(x) is less than or equal to c|x^2|. To do this, we need to find c, k, and x0 that satisfy the inequality.

1. First, let's simplify f(x) by dividing both the numerator and denominator by x^2 to get f(x) = (3/x - 5 + 4/x^2 - 8/(5x^3)) / (1 + 3/x - 1/x^2 - 7/x^5).
2. Next, we can ignore the lower order terms and focus on the dominant terms. As x approaches infinity, the dominant terms in the numerator and denominator are 3/x and 1/x^5, respectively.
3. So, we can rewrite the simplified f(x) as f(x) = (3/x) / (1 + 1/x^5).
4. To show that f(x) is in O(x^2), we need to prove that there exist positive constants c and k such that for all x larger than some value x0, the absolute value of f(x) is less than or equal to c|x^2|.
5. Let's choose x0 = 1 and k = 1. For all x > 1, we have 3/x < x^2 and (1 + 1/x^5) > 1. Therefore, we can write |f(x)| = |(3/x) / (1 + 1/x^5)| < |x^2 / 1| = |x^2|.
6. Thus, we have shown that for all x > 1, the absolute value of f(x) is less than or equal to c|x^2|, where c = 1. This proves that f(x) = 3x^3 - 5x^2 + 4x - 8 / (5x^5 + 3x^3 - x^2 - 7) is in O(x^2) for x ∈ R+.

To prove that f(x) is in Ω(x^2) for x ∈ R+, we need to show that there exist positive constants c and k such that for all x larger than some value x0, the absolute value of f(x) is greater than or equal to c|x^2|. Following a similar process as above, we can choose c = 1 and x0 = 1 to prove that f(x) is in Ω(x^2).