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1. (5 Points) Create the sign chart for f' (t) and use it to identify the local maxi- mum and local minimum values of f() 6t2 + 9t + 2(8 Points) Let f be a con...

Question

1. (5 Points) Create the sign chart for f' (t) and use it to identify the local maxi- mum and local minimum values of f() 6t2 + 9t + 2(8 Points) Let f be a continuous and differentiable function. Use the sign charts for f'(2) and f" (2) below to answer the following questions T(X)Find the intervals where is increasing and decreasing:b) Identify the location of any maximum O1 minimum values of f .Find the intervals where f is concave Up and concave down_Identify the location of any

1. (5 Points) Create the sign chart for f' (t) and use it to identify the local maxi- mum and local minimum values of f() 6t2 + 9t + 2 (8 Points) Let f be a continuous and differentiable function. Use the sign charts for f'(2) and f" (2) below to answer the following questions T(X) Find the intervals where is increasing and decreasing: b) Identify the location of any maximum O1 minimum values of f . Find the intervals where f is concave Up and concave down_ Identify the location of any inflection points of f _



Answers

$1-2$ Use the given graph of $f$ 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.

Well everyone. So today we're going to answer these questions here with our graph on the left. So let's find out where our function F increases and decreases. A good way to start doing this. Is to mark our maximum and minimum on the graph. So maximum and this is a minimum. So if we run our pencil down the graph here we'll find that this decreases to this minimum. From that minimum we increase to this maximum, we decrease the next minimum and then we increase for the rest of the graph. Mhm. Great. So if we draw some lines at Armand's and max is here we'll get A good idea of what x intervals to use for answers from 01 we are decreasing. X equals 02 X equals one. This looks like an X. Y pair but it's a range of X values. So from X equals 12 X equals three. We are increasing. Okay. Mhm. From X equals three X equals four. We are decreasing. And for Mexico's four to the end of our graphics equals six. We are increasing Great. So it's that out of the way let's look at where our function is concave upward and concave downward. So qualitatively here on the left I drew a little diagram of concave upward, concave downward. Um It used to be a little hard for me to remember which was which but if you think about it this this shape and concave upward is actually kind of shaped like a U. Like in the word upward. So if that helps you remember. So let's see where our function is concave upward and downward. So but here looks appreciate and here looks downward ish. And this part here actually is downward, it's curved, it's curved this way, Right like this many diagram I drew so this part is going to be downward as well. So this interval Hear from 4-6 is pretty easy to see because this is one big curve. Right? That's that. One is pretty easy to see. Let's write that down concave downward from 4 to 6 X equals four X equals six. And there's another pair here. So I'm going to leave space for that. Uh for this next part it might be easier to answer part 5 1st to find a point of inflection on our graph to remind ourselves, I wrote on the right point. Inflection is a point where punk function function, a point where the function changes from upward and downward or from downward to upward, cock a downward cop came up with all that. So it's basically where the direction of inflection changes and we can see that happening at X equals two. Right? Because this is concave downward and following this way is concave upwards. So we know that this is our point of inflection and in my notes I have it written as an Xy pair From 2-3. Great, awesome. This is to kind of three right here because this is one equals three. Okay, great. Okay. And just one more time we have to still answer 34 fully Number three, we know that this is a concave upward and if this is our point of inflection then we know that the concave upward nests stops about here. So this is going to be the interval uh X equals zero X equals two. And we know that this bout of concave downward Nous stops and X equals forks. That's when this next one starts. This big weird long one that goes to the end. So this is from X equals two. Let's arrest this. X equals to two. x equals four. And that should be it for a question. Uh Thank you very much for watching and I hope it helps

Alright everyone. So today we're going to answer these five questions here using this graph on the left. So let's find where function is increasing and decreasing. So if we start tracing up here we see that this is increasing. This is a maximum so it stops increasing. And we start going down here. This is a decrease to a local men. I think it's yeah local men and then we start going up this way increasing and this line the slope and exactly the spot is actually zero. It's kind of a cent tonic here. So there's no increase happening here. So I'm going to draw a little dot there. Look at that later as well, that's not increasing there. But as we keep going this way we see that increases similar. So let's look at our special points here max our men and that one little stagnant point there. And let's analyze what ex intervals we have going on here. So from X equals zero X equals once is an increase from 123. We see a decrease From 3-5. We see an increase in from 5-7. We see another increase because it's not like going down. It's a little weird to see something that's neither a maximum or minimum but you know it's still there, it's good to practice. So increasing this is X equals 02 X equals one. Okay this is. Uh huh. Thanks it cools 32 x equals five and then to X equal seven decreases our one. Little decrease. There's from X equals want to X equals three. Awesome. Good good good. So right now let's find where our function is. Concave upward and concave downwards. So just a little refresher. Have some notes here about what conclave operating concave downward mostly uh Qualitatively looked like. So let's see from our pictures here, if we look at this, this looks pretty concave downward. This here looks pretty concave upward. This looks pretty concave downward even though it's not like a full upside down. You you can tell this part here sort of points down and up here it actually looks a bit concave upward. So let's see and while we do this, we should probably answer number five first. We can find our points of inflection on our graph here. So just a quick reminder point of inflection is a point where the function changes from concave upward. Concave downward or from concave downward to concave upward. So it's basically where the directionality of your khan cavity changes if that makes sense. Let's just do orange or something. So you can tell that we start changing from concave downward and upward here about X equals to start changing from concave upward to downward of boys is actually cools score. We can tell that this is also a point of inflection because our contract dower concave upward changes here. Good. So we have our points of inflection. Let's call these are to to these are xy pairs by the way. These are not X intervals anymore for this question but three and four are going to be expendables again. Uh Mhm. for common. I'm going to call that three about And I'm gonna say that five Com 4. Okay great. So now we have these it's even easier to answer number three and 4. So we know that on these points of inflection are going to define where our intervals of upward and downward is our. So that's all right. So let's see we are concave downward from x equals 02 And recurrent cave downward from x equals 4- five. and likewise, we're concave upward from x equals 2-4 and we are Concave a bird from x equals 5-7, and I think that's about it for this problem. Thank you very much for tuning in, and I hope it helps.

So because we don't actually have the graph available, let's consider different possibilities. So if we have for example the graph X. I'm cubed -2X. For example, museum in here and look at this and we want to know intervals. How much is concave uh huh. How much is increasing? Well that's going to be increasing from here to here and that starts decreasing from here to here and it's increasing from there and not further. Then we want to know where it's decreasing. So we already defined that. Then we want to know um when it's concave uh So we see it's going to be concave up once it reaches this inflection point and it goes up like this. But then to be concave down to the left of that inflection point. So we see that the inflection point is going to be located right here because that's when the graph switches from concave down to con cuba. So that's our final answer.

So because we don't actually have the graph available, let's consider different possibilities. So if we have for example the graph X. I'm cubed -2X. For example, museum in here and look at this and we want to know intervals. How much is concave uh huh. How much is increasing? Well that's going to be increasing from here to here and that starts decreasing from here to here and it's increasing from there and not further. Then we want to know where it's decreasing. So we already defined that. Then we want to know um when it's concave uh So we see it's going to be concave up once it reaches this inflection point and it goes up like this. But then to be concave down to the left of that inflection point. So we see that the inflection point is going to be located right here because that's when the graph switches from concave down to con cuba. So that's our final answer.


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