a) Show that $ \frac{x^3 + 2x}{2x+1} \; is \; O(x^2) $
b) Find witnesses $ C \; and\; K $
My answer was :
$ x^3 + 2x \le c(x^2)(2x+1) $
$ x^3 \le c(x^2)(2x+1) , \; when \;c=1 , x>1 $
$ 2x \le c(x^2)(2x+1) , when \; c=1 , x>1 $
$ so \frac{x^3+2x}{2x+1} \le cx^2 , when \; c=2 \;and\; k=1 $
Is my answer correct ?
The order of your inequalities isn't very suggestive of what steps you are taking or why these inequalities might be true (how did you convince yourself that they were true?). But everything you wrote appears to be correct.