How do you defeat the last boss in sonic rush to save cream

If you are talking about the f zone boss: Robotnik will try to smash you with its arms, avoid them by running away or jump dashing the other direction! Kay, when he pauses while his arms are grounded, quickly jump attack the arms, Eggman will not be damaged but he will retreat to the end of the platform!
Soon Eggman will blast green waves from that end! Jump to avoid! Jump attack when he is finishes with his assault. Jump on top of his arm and run to his head and attack, but be careful, if you are too slow to run, will try to shake you off. Hold DOWN on the d pad when there is a warning sign above you.
After a few hits he will launch a death beam at you. It is very fast so keep jumping until he is done. He will also try to blast electric arcs, simply stand through the open parts.
At the end, Eggman will be very ticked and try to ram the platform, no worries, at the last strike of Robotnik's, jump on his and that will be the end for him. Congrats you beat Sonic's or Blaze's story