Use the given graph $ f $ to find the following.
(a) The open intervals on which $ f $ is increasing.
(b) The open intervals on which $ f $ is decreasing.
(c) The open intervals on which $ f $ is concave upward.
(d) The open intervals on which $ f $ is concave downward.
(e) The coordinates of the points of inflection.
(a) $f$ is increasing on (1,3) and (4,6)
(b) $f$ is decreasing on (0,1) and (3,4)
(c) $f$ is concave upward on (0,2)
(d) $f$ is concave downward on (2,4) and (4,6)
(e) The point of inflection is (2,3)
quick sketch of the graph that were given. And in this problem we're just going to answer some questions about the graph. So in part they were trying to find on what open intervals is our function or graph increasing. So I have some of the important points here in red and this first Redpoint is actually the minimum um or a local minimum of this graph. And we know that for f to be increasing, it needs to be going up and at a minimum value that's actually where we go from increasing to decreasing. If if we are actually going from increasing to decreasing, we could have a minimum value that's just like this and then the graph just stops, but in this case we're not just stopping, we're keeping going. So this is a minimum value local minimum. And whenever you have a local minimum value, that means that we're going from increasing to decreasing as long as it's like a Curve we're sorry, decreasing to increasing. So we're increasing from one to this maximum here from 1-3. So our interval for increasing for this first part that we're increasing is from 1-3. So you can say from 1 to 3 and I know that since our minimum value is when we go from decreasing increasing and this local maximum value is when we go from increasing to decreasing. So we're increasing in this interval between our minimum and or maximum and then the next part that we're, the next um interval in which we're increasing is going to be this part of our graph over here after we get to this point and then there's just a jump that seems to be going up and then our graph just stop. So in this, in this part of our graph from this point to this point We're still increasing throughout this whole range of values. So you can say that the other interval is from 4-6. And now for part B we're gonna try and figure out on what interval our function is decreasing. So yeah, decreasing is going to be this part of our graph here since we're going down and it's going to stop at this local minimum, since we know when we get to this minimum, we're going to be increasing afterwards. And then we're going to also have this interval over here. After this maximum down to this point here at for common one and then after the 10.4 common one you can see that our graph is increasing so that's the last interval of decreasing. So we have this first one from 0 to 1 And then we have the second interval from 3-4. Yeah. And now, right, that black for part C. Um we're trying to figure out where we are concave upwards, so it might be a little bit hard to tell on the graph that I drew but if you look at the original graph it was pretty pretty obvious at this point at two comma three right here is an inflection point and it's where we're going from concave up to concave down. So we're gonna have our graph be concave up from zero to this point at two, since this is our inflection point. And then pass this where concave down until we get to this point. And then at this point we do a jump and we're still actually concave down past this point. It it kind of just goes down here and then jumps up. It's not a very smooth curve. So there's no point where we're going from concave down to concave up. So The only place on our graph where actually concave up is this first interval from 0-2. And for part D we're gonna try and figure out where our graph is concave down. So as I said earlier, this point at two comma three is our inflection point. And so we're going from concave up to concave down at this point And from 2 to 4 we're still concave down And then from 4 to 6 were also con Cape town. So our intervals are going to be from 2 to 4 or this part of our graph here And then also from 4 to 6. So our two intervals from 2 to 4 And from 4 to 6. And then the last thing that we're gonna find on our graph is the actual inflection points of our graph. And so as I stated earlier this this point is actually right here. Two comma three. And the reason I can tell that it's an inflection point is because we're going from being concave up to concave down inflection points, Our points in which we're either going from concave up to concave down or from concave down to concave up. So this point here at 23 is the only point on our graph that actually does this. At this .41, we're still we're going from concave down to concave down again, so it's not an inflection point. So there are only inflection point is at this 0.0.2 comma three.