Growth of Functions

Need solutions for the following questions with step by step work shown
(a)
Let $f(x) = x^2 +x + 15$ and $g(x) = x^2 log(x) + 10$. Prove that f(x) is O(g(x)) by:
1. Providing witnesses C and k and
2. Proving that the inequality holds for the value you choose in 1.

(b)
Let $f(x) = x^5 + 10$ and $g(x) = x^5 + x + 10$
1. Prove that $f is O(g)$
2. Prove that $f is ?(g)$
3. Given parts a and b, what other relationship can you show about f and g?

Discrete Mathematics Functions
Jack Smith Jack Smith
Matchmaticians Growth of Functions File #1 File #1 (PNG)
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  • Jack Smith Jack Smith

    Please provide full answers, and please solve them in a way that a someone who semi-understands (me) can comprehend and study it in the future

  • Nirenberg Nirenberg

    That's what Matchmaticins actually expects from the answerers. All solutions should be written so someone with basic background can understand. I wrote a very detailed solution, but let me know if you need any clarification.

Answer

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Nirenberg Nirenberg
  • Nirenberg Nirenberg

    Leave a comment if you need any clarifications.

  • Jack Smith Jack Smith

    thanks phil. Can you please label parts 1 and 2 of (a) and also if possible solve the question with some other relatively lower value for C (say 3,4,5,17) .

  • Jack Smith Jack Smith

    Will really appreciate if you'd write back:)

  • Nirenberg Nirenberg

    What I have written for (a) answers part 2 of the question. The last line answers part 1, i.e. k=1 and C=e^2.

  • Nirenberg Nirenberg

    So the answer for part 1 is k=1 and C=e^2.

  • Nirenberg Nirenberg

    In this kind of questions values of C does not matter, you just need to show that for some C and K the inequality holds and that's what we have done here.

  • Jack Smith Jack Smith

    This is actually a homework question for a course I am already on thin ice on. The grader is kinda sadistic and I am pretty sure he won't give me any points unless I solve the first part(C and K) followed by the second (inequality).

  • Jack Smith Jack Smith

    Please understand my plight and sorry I am coming across as annoying.

  • Nirenberg Nirenberg

    Sir, the answer for part 1 i: s k=1 and C=e^2. The answer for part 2 is in the file I uploaded. You seem to be overthinking this. Everything is ok, relax and submit your homework.

The answer is accepted.