Suppose the derivative of a function $ f $ is $ f'(x) = (x + 1)^2 (x - 3)^5 (x - 6)^4 $. On what interval is $ f $ increasing?
Okay, so we're told that the derivative FSF private bank and it's written right here and we're being ass on one into was f increase. So what we are going to do is we're going to look at So this does many different ways to do this. For some, it may be obvious for some, not B. I'm going to try to make it as obvious as possible. So we see that we have, ah, square A to the fifth and to the fourth. And so we know that any number squared were always deposited in any number two. The fourth will always also deposited. But this will not always be positive, so we can do this. Fifteen tells us that is being a small flying ex ministry five times X minus three times X minus three times minus three times X minus two times expensive. So what? I'm going to do it. I'm gonna pull one of those x minus three out and it's going to give us prime of eggs equals X plus one. So these are going to remain the same. That was one squared times X minus three to the forint. Times X minus six. Tune in for it times X minus two And so now it should be a little bit more clear. All the least three terms are always positive, so they're going to be always greater than do that. They're always going to be a greater than Teo positive. And so that means the sign of the function of prime is determined by this guy. Wrote your ex ministry. So when three minutes when X is greater than three, that is only appointed in which justice term right here is positive. So that means the sign Ah, whenever the sign of f promise positive. That's when efforts increasing, so it is increasing from three to infinity.