Question
IonnucJnswn thc follawing questions= nasedeiven Fraph of k(r)[3 pts] On what intervals or at what polnts k(x) concave down?[3 pte] Lercribe wynat happening to K' (x) when k(x) concave novnaEEe]# E(-) Is che Tunction xhose derivatlve k(x); where would K(x) be horizontal?
Ionnuc Jnswn thc follawing questions= nased eiven Fraph of k(r) [3 pts] On what intervals or at what polnts k(x) concave down? [3 pte] Lercribe wynat happening to K' (x) when k(x) concave novna EEe]# E(-) Is che Tunction xhose derivatlve k(x); where would K(x) be horizontal?
Answers
Determine the interval(s) on which the function is concave up and concave down. $$ k(x)=-3 \sqrt{x}-1 $$
Was she? Even this therefore mixes. It'll suit he despite X into expensive way they for from a weapon down and then also reporting a confession for they would you be differentiating the next time? Let's difference. It'd first time. So, does that actually question Mr Politics? The position of the storm breaks in municipal elections really missed me. Okay, let's see. This one doesn't confrontational. What is it? Defied. The first of your give me everything it's disturb are actually experience. It's just changing. Next time is there for the physician off. Mr Parks will be as it is, so there could be minus two less used to residents and the position of experience to you. I used to like this one second. Well, yeah. So finally infection. Really great that the radiator because of generating acceptable super. Okay, so there is a point. Okay, we understand they double in which the function is one care of a known for their baby. I mean, our sign it science off the gruesome extricating 01 society week between Boston discriminated. So from here, you can say that affects yes, come care for. There's strong more for me. And I think this? Who cares down for next storm? Anything for you? Well, certainly. Part of infection. Your water to go. That is that is that
In this problem, we want to find where the graph of the function is conquered up. The rushes down. As well as the inflection points, I have X is equal to each of the three x minus nine GTX. This question is showing their understanding of how to use the second derivative to analyze on cavity. Inflection points for a function. We proceed to the steps listed here to solve so first and foremost we have to use equitable prime. Therefore we need to find after the crime. S prime is 3 to 3 X minus I need to be acceptable. Prime is 93 X minus 90 X are getting points occur whatever vegetable prime equals zero. So 90 to the extent of the two, X minus one equals zero, gives either two, X equals one or X equals zero. Definitely evaluate on the sign of a double prime. Now on a sign chart for all interval separated by repetition points. So from negative year zero after the private negative from zero to infinity it's positive. Those, we conclude the cavity is concave up on 02 infinity. Concave down on negative 32 0. Since inflection points occur welcome, cavity changes sign. It's therefore means that X equals zero is the inflection point.
We're starting with a function of E to the X Times X minus three and we are determining where this is. Khan gave up. Khan came down as well. It's the inflection points to look at Cannes cavity. We need to get to the second derivative. The first derivative is going to use the product rule. We have first term times the derivative the second, which is one plus the second term, which is the X minus three times the derivative of the first, which is either the axe and I can simplify that down some that would give me E to the X plus X E to the X minus three e to the X, and I can combine the like terms of either the X minus three to the X, so I really have negative to e to the x plus x times e to the axe. My second derivative derivative of the negative to e to the X is negative two e to the axe and the derivative of the X eat of the axe will use the product room first times the derivative the second, which is e to the X plus the second term eat of the X Times, the derivative of the first, which is one. We will get possible inflection points any time. The second derivative is non differential. Our it is equal to zero. It is never non differential, so let's just check to see where the second derivative is equal to zero. I'll call this a P I p for possible inflection points. I can't simplify it down a little bit more to if I want to, because negative to eat of the X plus either the X is the Samos negative e to the X. Okay, so let's see where this equals zero. I can factor out and eat of the X, so that would give me e to the X Times negative one plus X equals zero. Now that's gonna be true. If either eats of the ax equals zero or negative one plus X equals zero. Even the axes never zero because eat any power is a positive answer. So the only possibility is going to be if negative one plus X is equal to zero. And that's just a little one step equation. If I add one, I get an answer of X equals one. So let's check around that value to see what happens to the sign of our second derivative OPIC values of zero. And to now remember, I'm filling this into the second derivative. If I feel zero in, I will get negative one plus zero. A negative answer. If I felt to in, I will get a positive answer. So we have it going from con cave down. You can't give up. So our function is con cave down from negative infinity 21 It is con cave up from one to infinity And there is an inflection point when exits one to find the function Coordinate For that I would go back to the original function and fill one in for X. If I do that, I get a function value of negative too A
All right. In this problem we want to determine with a function F of X is concrete? Up versus down. And to find the point of inflection F of X equals X to the negative three X. This question challenges our understanding of application of the second derivative. So it's all we need a procedure steps one through five less than the outline here. Is that 1? We calculate the second derivative F. Prime is even negative three X one X three X three X. So after prime is simply 1960 to 93 expose nine X. Negative three X. The partition going selectable primary equals zero or undefined. Thus we have sending our noble practical 09 X minus six equals zero X equals two thirds the size of the vegetable prime. We evaluate the sign of a possible crime on all intervals separated at the institution points from negative three and two thirds of the crime is negative from two thirds to infinity. Ethical prime is positive. This map directly undercut cavity for which we have access content up on two thirds to infinity. Khan came down on negative 32 3rd. Finally, the inflection point occurs wherever active crime changes. Science is defined, Thus x equals 2/3 are inflections.