I think we'll be fine. I base this on times in the past where people felt like they might become obsolete and turned out they could just do other things. I also think if this happens it will be in a longer timeline, beyond 10 years. It might just be gradual enough that not everybody has nothing to do at the same time. Besides, what motives do the robots have to even do anything? We are programmed for survival, most of the things we do or want have to do with prolonging our own lifespan, which in the end will still result in death. Life is simply defying death until you have at least ensured the survival of your species. Our surroundings change faster than we change, so we do stupid things that actually encourage our dead, but they still work with the same principles that we needed in the past to improve the survival of the species. You would need to put all of these things into a robot and find mechanisms that make it unable to deviate from this or else it is a whole lot of effort for nothing. If you have no urge to survive or procreate that is everlasting it will come to an end sooner rather than later.