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X - 0 1 2 | 3 | 4 flx) 5 | 2 2 6 | 8 10 f'(x) 8 | 5 -3 -1 | 0 -3 f"(x) 6 3 0 -3 /-2 4 The table above gives the output value; the derivative value; and th...

Question

X - 0 1 2 | 3 | 4 flx) 5 | 2 2 6 | 8 10 f'(x) 8 | 5 -3 -1 | 0 -3 f"(x) 6 3 0 -3 /-2 4 The table above gives the output value; the derivative value; and the second derivative value of the function f for specific input valuesQuestion 2 (1 point) Give one X value for which_ f(x) is increasingQuestion 3 (1 point) Give one X value for whichf (x) is concave up.Question 4 (1 point) Give one X value for whichf (x) has an inflection point:

X - 0 1 2 | 3 | 4 flx) 5 | 2 2 6 | 8 10 f'(x) 8 | 5 -3 -1 | 0 -3 f"(x) 6 3 0 -3 /-2 4 The table above gives the output value; the derivative value; and the second derivative value of the function f for specific input values Question 2 (1 point) Give one X value for which_ f(x) is increasing Question 3 (1 point) Give one X value for whichf (x) is concave up. Question 4 (1 point) Give one X value for whichf (x) has an inflection point:



Answers

The first and second derivatives of the function $f(x)$ have the values given in Table 1.
(a) Find the $x$ -coordinates of all relative extreme points.
(b) Find the $x$ -coordinates of all inflection points.
$$\text { Table 1 Values of the First Two Derivatives of a Function }$$
$$\begin{array}{ccc}
\hline x & f^{\prime}(x) & f^{\prime \prime}(x) \\
\hline 0 \leq x < 2 & \text { Positive } & \text { Negative } \\
2 & 0 & \text { Negative } \\
2 < x < 3 & \text { Negative } & \text { Negative } \\
3 & \text { Negative } & 0 \\
3 < x < 4 & \text { Negative } & \text { Positive } \\
4 & 0 & 0 \\
4 < x \leq 6 & \text { Negative } & \text { Negative } \\
\hline
\end{array}$$

Question 25 wants you to determine where the function F of X equals four X cubed minus six X squared plus two X plus one is concave up or down and list any inflection points. So to do so, we need to get to the second derivative. So F prime of X is 12 X squared minus 12 X plus two F double prime of X is equal to 24 x minus 12. So first we can find inflection points. I would be when f double prime of zero or of X is equal to zero. So setting your equation 24 x minus 12 equal to zero gives you X is equal to one half. Plugging that into FX F of one half is equal to once or inflection point is at one half one. Now, testing work on cavity is so testing for X. Less than one half take F double prime of a number like zero that gives you negative 12, which means that when X is less than one half, you're concave down and testing a number greater than one half so F double prime of one is equal to 12. Therefore, when access greater than one half, you are concave up, and those are your answers for question 25

Grateful. We want to find where the function at this conclave. Up versus down exactly equal to X. To the fourth minus six xq minus 24 X squared plus three X plus one. As the outline on the left suggests we're gonna be using the second derivative. Have to solve this problem. First we find out the prime. Then we find this critical numbers where after prime zero or under fine. Then we use the signing charged with designing export crying and all interval separated critical numbers. Except for we can conclude where after I can't keep up with concrete down version are signing chart as well as determine the inflection points. So ethical prime is taking to derivatives 12 X squared minus 36 X minus 48 or 12 times X squared minus three X minus four. Factoring we have zeros at negative +14 These are critical numbers to the left of negative one. After the crime is positive between negative one and four. After what time is negative and to the right up for after the prime is positive. Thus, efforts can't give up on negative 31 for the infinity can keep down a negative 1 to 4, and we have inflection points at negative one and four where the con cavity is changing.

Question 23 would like you to determine where f of X equals negative three X squared plus two. Is concave up or down? Uh, and if it has any inflection points, so to do so, we need to find the second derivative. So F prime of X is equal to negative. Six X. If double prime of X is equal to negative six because this is less than zero, that tells us where concave down and the only time there's an inflection point is when I have double Prime Vector is equal to zero. Therefore, we are concave down always because there is no inflection point. That's your answer to question 23.

So how do we determine the intervals? On a fraction are came up and complete that If you look on the right hand side, we know that when the second derivative of that function is greater than zero is calm came up and cc just danceable concave. Similarly, when the function is taken to the second riveted and it is the lessons about is concave down. And lastly, this point is always good to know when the second derivative of a function is equal to zero. That X value is the point of inflection. Soapy Oi, there, just Ansel point of inflection. Okay, so you know what we have to do? We need to find a second derivative of a function called the left hand side X to the power four minus two x cubed plus one. So you take the first line which gives us three times a panel by the base full X and B Subtract one from the power. Do the same here. So we do free times later to just like to six x and then we take away one from three just to, and the first rooted off one is zero. So we don't eat that now we take the second moved to so we just differentiated again. We do the same thing. So times the power by the base four times free is 12 x three minus one is to go to that and we do the sinking here tee times. The negative six is a troll and it was won by the power. So we found the second derivative. So what we need to do now is find the point of inflection because that would help us find the intervals that can keep up and compete down. So we know that current off inflection is when f x f dash dash X is equal to is up. So we need to do so. Second derivative here 12 x squared minus 12 x Teoh equal at zero. Look on. We can do some fact arise appear to make it a lot easier. We can take up 12 and X from both from both expressions and over here we end up with X, and here we end up with one. So this means that 12 x x equals O and X minus one. The sequel Go meaning X equals ill For this one, we just divide 12 on both sides. And for the bottom one, X equals one. It's that one on both sides. Okay, so now we have Ah, uh, we have a point of inflection. So now that we have this, we cannot establish that they were going to be three intervals. Okay, 1st 1 is from negative. Infinity to zero. So now you to affinity 20 a second interview. He's going to be values between zero and one. I know Third Interview is going to be values greater than one. So between one and infinity. So first interval is for values lessons about so that means or numbers lessons up So negative 123 ordering up to negative infinity. So that's our first interval. Second interval between zero and one on a 34 for all values greater than one. So 2345 would react to infinity. So I third interview is between one infinity Now in order to decide which includes our Khan came up and concave down. We simply need to do is take a number from each interval and substituted into our secondary to function. So for the 1st 1 So the interval between negative. Infinity and zero. We can always take negative one exoplanets between eight and 15 0 So we get a negative one on the subway in tow, R F dash dash X, which is 12 x squared, minus 12 X. So So I'll just write first here. So we have 12 times minus one squared minus 12 times like this one. So minus one squared is just 11 times four is 12 and minus one times minus 12 is plus 12. So you have 24 which is greater than zero. Which means if you look at the talk right when F Dash Dash X is greater than zero is Khan came up. So this means our first interview is corn cave up. And remember, CC just stands will come cape. So let's take all second interval. So values between zero and one so you can pick isn't quite five. Um, so that substitutes or great five in. So we have 12 times. Okay, five square minus 12 times. Dont pay a fine now is OK, but you know that five squared is just 25. So, using that same logic, you can just realised that his Oakland five squared is so quick to five or ah, half squared is 1/4. Okay, so we know that 0.5 squared is looking to five. So that's basically 1/4. Okay, so this is basically 12 times quarter on 12 times 1/4 is basically 12 divided by four. So 12 divided by four is three. And we know that when you times buys, okay, five who are coughing. So we need to do negative 12 divided by two, which is six. So negative 12 divided by two years. Negative. Six here. And so we get three minus six, which is virus free from you. See that this is less than zero. When you look at our inequalities, we know that when f dash dash x is less than zero is complete down. So over here we can say it is concave down and for third in through between one and infinity weaken to speak to, I'll be substituted into a f dash dash x. So we have 12 times to you squared minus 12 times to we know two squared is four on four times 12 is 48 minus 12 times two is 24. So This gives us 24 which is greater than zero. And we see we're Miller Kyle. Equality When f dash that X is greater than zero. It's called Click Lock. So you know that our third interval is concave. Oh, So our first interval between negative infinity and set up its conch A cup second interval between zero and one concrete down third into between one and infinity complete up. And we know that all points of reflection when X equals zero and X equals one so and we have it.


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