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Consider the function in the graph below:Graph the derivative of this function.Over which intervals of the domain on this graph will the second derivative be positi...

Question

Consider the function in the graph below:Graph the derivative of this function.Over which intervals of the domain on this graph will the second derivative be positive?

Consider the function in the graph below: Graph the derivative of this function. Over which intervals of the domain on this graph will the second derivative be positive?



Answers

THINK ABOUT IT Sketch the graph of a function whose derivative is always positive.

For the following problem, we want to sketch a function whose 1st and 2nd derivatives are always negative. So that means it's always going to be decreasing, it's always going to be concave down. So what that's gonna look like, for example, as if we have each of the X. But we wanted to be decreasing in concave down. Um That will look like this. Now, you see the function is decreasing and it's concave down. Um And then we can also uh consider the function who's 1st and 2nd derivatives are always negative or whose whose first derivative is negative, but the second derivative is positive. So it's going to be concave up but decreasing. So it's going to be something like this can't give up decreasing, so have that is concave up but decreasing. So that's our final answer.

All right, So this is chapter two section for problem number 12. Um, what we have here, we have this given table, the list off a bunch of points, the form of function, and I have plotted those points off to the right. And what we're looking for is the first derivative of this function. You want to determine whether it's positive or negative, and we also want to determine whether the second derivative it's positive or negative. Um, and you can do it only looking at these numbers, but I think that for the sake of this video, you know, beam went much easier to visualize with this graph, um, that you can sort of connect these dots to get a sense of what the ah function could potentially look like. I mean, it has the could do something weird like this. It's not guaranteed that it doesn't, but we can get a general sense of where it goes. Um, and with this we can tell that the function is always decreasing. It's always going down, which means that the slope is always is always negative and that the first there is it is, therefore is also always negative um is the function never goes up, just goes down. We can also tell that it is always con cave up. It's not a huge concave ity, but it's there, as opposed to something like this, which is Kong cave down. This here would be a negative second derivative. Well, con cave up is the first ER is a positive second derivative because, um, the slope is always increasing. It's always going up, and here it's a negative slope with the negatives are getting closer and closer to the positives as it goes down. Um, which means that the second derivative is positive. Do we have first?

All right, this is Chapter two, Section four. Problem number 10. Um, for this one, we have the graph of f of X over to the left, the green squiggly line, and we want to determine the intervals when the first derivative is positive and the interval when it's negative and the same for the second derivative. So if we look at the first derivative first, um, when it's when the first derivative is positive, is when it has a positive slope and it's negative when it has a negative slope. So we want to do is we want to look for the points where the slope changes from positive to negative or from negative to positive. So that's basically on the maximums of the graph for the minimums when it starts going from up to down or down to up. So there's one there. There's one somewhere around there and one right there. And this problem we only have to estimate since the graph isn't isn't precise. It only is this skill from 1 to 4, Um, but here we can tell that there's a positive interval right there. Positive interval right there, and a negative interval right there. and right there. So if we want to mark these numbers, this looks like it's about at X equals 0.5. Ah, this 2nd 1 looks maybe 0.8, and the 3rd 1 is about our site, Not your 0.8 is 0.8 from from here. So 1.8, then for this last one about 3.5, um so is positive from that's theirs are zero. So that means positive from zero to 0.5. Then we have the union symbol right here, which means there's a gap between them and it starts up again at 0.8 you 3.5. So there's are positive intervals from the first derivative for a negative intervals that starts off right here at 0.5 and goes until 1.8, then picks up again ahead 3.5 and goes until four. So that's what we do for the first derivative. Now, for the second derivative, it's sort of a similar process instead of except instead of looking at slope, we're looking at the direction that the line is curving. So here the line is curving downwards, which means the second derivative is negative here. It's curving upwards, which means it's positive, and there it's negative again. So here we want to try and find the the points where the curve switches from curving downward to covering upward or switches from curving upward to curving downward. And this is probably just a little bit more difficult to see than these nice, clearly defined peaks right there. Because, like especially if you get into something like that like it could be there like that could be a downward curve and starts curving upwards there. Or, you know, it's it's slightly more difficult, but I think we can still do this. So there's clearly switches somewhere around there from a negative Ah, negative second derivative to a positive second derivative. And then somewhere around this three mark, it's which is again, back to negative. So here are negative intervals from zero to about one. Then it picks up again at about three and keeps on going negative until four. And then here we just have this one positive interval, going from 123 and so there's our answers. Um, it's a bit messy, but again first, for the first derivative positive from 0 to 0.5 and again from 1.8 to 3.5. It's negative from 0.5 to 1.8 and again from 3.5 to 4. And for the second derivative of positive between one and three. And it's negative between zero and one and again between three in four. So there's your answer toe problem Number 10 Good bye.

All right, this is Chapter two section for problem number 11 and what we want to do here is we have this graph of f of X over to the left and we want to determine the interval that the first derivative is positive and the interval that it's negative. And we also want to determine the interval when the second derivative is positive in the interval when it's negative. So if we start off with the first derivative, um, positive when the slope is positive and it's negative when the slope is negative. So we're looking for the points when the slope changes from positive to negative or from negative to positive home. And so this just means when the graph tops going up and starts going down or stops going down and starts going up. Um, and we won here in this case. That's when the graph stops going up and starts going down right there. I'd say that's about negative 0.5. Um and we don't have any other changes. That's all. Negative there. Sorry. It's all positive here. Hands. The derivative is all negative on the right side of that. So the positive interval is from negative, about 2.3 to negative 0.5. And it's negative from negatives, your 0.5 until four. So that's it for the first derivative. The second derivative is slightly trickier. We do basically the same thing, except instead of looking at Slope, we're looking at the curve of the line. So when it's curving downwards, it's a negative second derivative when it's curving upwards with the positive second derivative. Like right here, it's dreamt Lee curving upwards. So that's positive. Second derivative. Then somewhere around that point it changes, and it's a negative second derivative over there because it's it starts going up in its in occurs downward. Um, and I would call this point about zero point five, which means that are positive. Interval for the second derivative is from 0.5 24 and our negative interval for the second derivative is from negative 2.3 to 0.5. And there you go again, the first derivative. It's positive from negative 2.3 to negative 0.5 and negative from negative 0.5 to 4, and the second derivative is positive from 0.5 to 4 and negative from negative 2.3 to 0.5. So there you go, Goodbye


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