(a) Graph the function
(b) Use l'Hospital's Rule to explain the behavior as $ x \to 0 $.
(c) Estimate the minimum value and intervals of concavity.
Then use calculus to find the exact values.
$ f(x) = x^2 \ln x $
so we want to grab the function. F of X is equal to x word town. Natural market back. We want to you both a tolerable to explain the behavior as X approaches zero and we want to estimate the minimum value and intervals of Kong cavities using the graph and then use calculus to find the exact values. So I went ahead and wanted a graph of next year. So the first thing they told us to do cluster graph is which where he did. Now let's go ahead and be vulnerable to explain the behavior as X approaches there. So since this function is undefined on the left, we're gonna need to look at in behavior to the right. So we want to look at the limit as Ex Purchase zero from it, right of X squared, natural, long of X. And now, if we were to just go ahead and plug zero from the right X squared goes to zero and natural longer base close to nature, did he so that we want to use will be told you were going to rewrite this so it's zero over zero or thin the over. So to do that we could go ahead and write it as natural log of X over one over next square. And now that we were to go ahead, evaluate, This would be a negative entity over infinity, which tells us we can apply locals. All right, so let's go ahead and apply local social. So now this will be equal by local polls rule, too, as extrapolate resume from the fight of so the derivative of our numerator, which is going to be one of Rex over the derivative of our denominator, which is going to be negative too. Excuse me. Now we can use a little bit of algebra to rewrite this as limit has expect pictures from the right, Uh, negative, too, X. Where, And we know that this year will approach zilla and that we look at the graph that does explain the behavior of this because it does look like it's getting closer and closer to zero at this point Now, for Part C wanted to find our minimum value and so just kind of go into the graph here. That point I have on the ground looks like about what our minimum should be. And then we also wanted Check for change Akane cavity and it looks like of our home. 0.2. We have a change of con cavity, so this would be 02 about 0.2. It looks like the function is calm Kate down. And then from 0.2 to infinity, it's calm Cave. Oh, right now, let's go ahead and you do a bit of capitalist too. Actually find what are minimum and calm. Cavity interval should be so let's just write down against the F of X is equal to x squared times the natural Uncle Lex So first derivative will be so prime. Uh, ex want to take the derivative here but after his product Cool. So it's the first function times the derivative of the seconds by the X of natural architects and then plus in the opposite order. So natural objects times dreamt of respect acts of exclaimed So the derivative of natural log of exit La Brix, the derivative of experience is to X and now we can go ahead and simplify this to you. X plus two X central log of X. Now, to find our critical values, we want to set this equal to zero. And actually, before we do that, let's go them factor this. So I'm gonna pull out an ex. So ex times one plus two natural log of X. Now, let's go ahead and set the secret here. So this tells us either ex busy with zero for, um, one plus two. Natural log of X is equal to zero, so we can't charge one divide by two. And that would give us natural Long of X is equal to negative 1/2 exponentially eight inside. So we get X is equal to e to the negative 1/2. So it didn't look like our minimum value was here and exited with zero, so we could just go ahead and a lot of that one. And since I was also on the edge of our domain, we would know that that value there should be included anyways, so our minimum occurs at E to the negative one. Um, and let's just go ahead and plug that in to see what we get. So of e to the negative 1/2 is going to equal to e to the negative 1/2 squared times natural OD to the negative one so need to negative one house squared is just gonna become e to the negative First power and then the natural are gonna be here. Cancel out. We're luck with negative one. So we get negative one over two. So this here would be our minimum value at X is equal to need to the packet of what happened next. Let's go ahead and find our, um, comma cavity interval. So a double prime prime was done right? At first again, a final X was equal to x times to natural log of X plus one. So now, to find a double time, well, go ahead to use product rule again. So So we write the 2nd 1 take the derivative of the first and then add them in the office door. So ex the by the X to natural log of X plus one. So the derivative of X is going to be one, and the derivative of two natural long X is going to be to overextend. The derivative of one is just want to be so we can go ahead and rewrite this a to natural log of X plus one and then the exes over their counsel to were just up with plus two. So this is going to be too natural. Log of X plus three. Now, let's go ahead and set this here. And that's going to tell us that you will get so subtract the three. Divide by two will get natural. Log of X is negative. Negative Three, huh? And exponential hating. Each side will get Xnegative you need to the negative three house. So now we would just go ahead and use this for interval. So we knew it looked like we come look over here again. It looked like it was conquered down to start. So from zero to E to the negative three house who say this is going to be con que down. And then from E to the negative three half's to infinity is going to be conned up. It looks like so we ended up getting our exact interval for Kong cable concrete down as well as our exact minimum