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(5 pt) Use the information provided to sketch the graph of the function_f(-1)=0,f()=0,f(3)=0 f(-2)=10,f(2)=6 extrema at ~0.5,-1) point = of inflection at (0.5,2) in...

Question

(5 pt) Use the information provided to sketch the graph of the function_f(-1)=0,f()=0,f(3)=0 f(-2)=10,f(2)=6 extrema at ~0.5,-1) point = of inflection at (0.5,2) increases for (-0.5,2) decreases for 5', ~0.5)and (2,0o) concave up for (20,0.5) concave down for(0.5,02)

(5 pt) Use the information provided to sketch the graph of the function_ f(-1)=0,f()=0,f(3)=0 f(-2)=10,f(2)=6 extrema at ~0.5,-1) point = of inflection at (0.5,2) increases for (-0.5,2) decreases for 5', ~0.5)and (2,0o) concave up for (20,0.5) concave down for(0.5,02)



Answers

Sketch the graph of each function. List the coordinates of where extrema or points of inflection occur. State where the function is increasing or decreasing, as well as where it is concave up or concave down $f(x)=2 x^{3}-3 x^{2}-36 x+28$

We need to find the extreme, um, the points of inflection where the function is increasing or decreasing and where it is con cave up on con cave down of the function five X cubed minus three X to the fifth. So let's start with the extreme. Um, so we find our first derivative and set it equal to zero. Our first derivative is 15 x squared minus 15 x to the fourth. This gives us that X equals negative 10 and positive one. So our coordinates are negative one. When we place into the function, we get negative too. 00 and one positive, too. Now we need to find where the second derivative is equal to zero. To find the inflection points so set in the second derivative, which is 30 x minus 60 x cute, equal to zero. We get that X equals zero X equals negative 00.707 and X equals positive 0.707 So that means this 0.0 is an inflection point. We also have an inflection point at negative 0.707 It's why coordinate is negative 1.237 and we have another inflection point at positive 0.0.707 It's why value is positive. 1.237 where our first derivative is equal to zero. Sorry, we're first derivative is greater than zero. Tells us where our function is increasing so we can see the function is decreasing when access between negative infinity and negative one it is increasing when x is between negative one and one and then it is decreasing again from one to infinity where the second derivative is greater than zero tells us where function is con cape up Sorry function is con cave up in the intervals Negative infinity to negative 0.707 It is also con cave up from zero to positive 00.707 It is con cave down from negative 0.707 to 0 and also from positive 0.707 to infinity. Using the information we just found. This is how we would sketch our function We can see that we have a minimum at negative one negative too a maximum at 12 We have an inflection point at negative 0.0 Zoo Sorry negative 0.707 and positive 0.707 It is increasing in decreasing in the appropriate regions and it is con cave up in Khan Cave down where it should be.

Okay, so we solve this problem by first finding the 1st and 2nd derivative. So the first derivative is equal to negative 4/3 X minus four to the negative one third power. And the second derivative is equal to negative 4/9 X minus four to the negative four thirds power. Well, you can see that the second derivative is always greater than zero. It's never equal to zero. So there are no inflection points. We can see from the first derivative that we will have a maximum at X equals four. So on the function we have the 40.4 comma five since the second derivative is always greater than zero. This tells us that our function is con cave up from negative infinity to four and from Ford to infinity. Looking at the first derivative, we see that the function the first derivative is greater than zero when X is between negative infinity and four. So this tells us that our function is increasing and it is decreasing from forward to infinity. Since the first derivative is less than zero, this is what the sketch will look like. It has no inflection points. It has a maximum at 45 It is always con cave up and it is increasing in decreasing on the appropriate regions.

And this problem, I want to take a look at this function and its graph and just look at if it's increasing or decreasing and over what intervals and if it's concave up or concave down. So I'm gonna start on the left and just kind of worked my way to the right. It looks like this piece here is all the same. So it looks like that's from about negative infinity. Up to See one. Well, that's it too. Huh? And from negative infinity to to it's increasing and it's concave down. Okay, I'm gonna switch colors and then look at this part here and really it looks like it switches. So I'm gonna go to like there. So from two to about one, it's now decreasing and it's still concave down and about at that point, right there it switches. So at about 1- zero it's still decreasing. But it changes to concave up and then let's see what else we got here. This piece here, right? From zero to what looks like infinity is increasing and is concave up

All right. We want to determine where the function F of X is concrete, up versus down and to find the point of inflection affects is equal to five. Experiment is actually four. Are the outline below. Hence that we use the second derivative to understand this question. The second derivative or rather the sign of the second derivative at the point of the function tells us about the con cavity. Accordingly. We can follow through steps 1 to 5 to solve. So first step one, we find the derivatives prime is 10 X minus sports cube at double prime is 10 minutes 12 X squared. Is that too? We find the partition points of the prime. That's where it's equal to zero undefined. Thus we have two times five minutes six X squared equals zero. Giving example social minus 506. Next we take the sign of vegetable prime on all interval separated by a partition points. So from negative three negative 5/6, negative 5/65 over 65/60 infinity after the prime is negative positive negative, respectively. Which means that if it came up on negative five or 65 or six, concrete down negative negative five or 635 or 16 30. Finally, the inflection points occur wherever the the cavity left changes sign. Thus, we conclude that executes cluster management 5/6 RR inflection points.


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