The Big O Notation and the Thetha Notation

Question

I was instructed to find whether $$x*⌈x⌉*⌊x⌋$$ is$$ O(x^3) $$ or $$Big Thetha(x^3)$$ I tried to do a solution by cases, and i got : if x is not an integer, $$x=b+є$$ $$⌈x⌉=b+1$$ $$⌊x⌋=b$$ Then $$x*⌈x⌉*⌊x⌋=(b+1)(b)(b+є)$$ If x is an integer , then $$x*⌈x⌉*⌊x⌋=x^3$$ How can i continue with my proof. Thank you.

Answer 1

Observe that $\lceil x \rceil < 2x$ and $\lfloor x \rfloor \leq x$. So your entire expression is bounded above by $2x^3$.