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(a) Find the intervals on which $ f $ is increasing or decreasing.

(b) Find the local maximum and minimum values of $ f $.

(c) Find the intervals of concavity and the inflection points.

$ f(x) = x^2 \ln x $

a) decreasing: $\left(0, \frac{1}{\sqrt{e}}\right) \quad$ increasing: $\left(\frac{1}{\sqrt{e}}, \infty\right)$

b) Local Minimum at $x=e^{-1 / 2} .$ The point occurs at $\left(e^{-\frac{1}{2}},-\frac{1}{2 e}\right)$

c) Inflection point at $x=e^{-3 / 2},$ concave down on $\left(0, e^{-3 / 2}\right),$ concave up on

$\left(e^{-3 / 2}, \infty\right)$

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Missouri State University

Campbell University

Oregon State University

Harvey Mudd College

So we're being asked to find the intervals in which f is increasing or decreasing on. Then we're being asked to find a local maximum and into other contracting inflection point for Africa. So the before we perceive with this problem we have Tio take note of a couple of things I've been taking out of something since it is ever vex IQ with expert terms. Island of X. We know. I think it is very important to consider the domain of dysfunction just in general, whenever you're doing these kind of problems, it was really a good idea to just get a mental idea what the domain it. And since we're looking at Allen of X, we know that the domain for Eleanor Rex is all greater than zero. Um, so this is your domain, because any number the Ellen function cannot equal do, and it doesn't accommodate negative value because the grass of something like this Ah, kind of. And so that means we're evaluating a point and looking at our interval or where the function of increasing and decreasing, you know, look at what that is greater than here. So speaking of finding values were increasing decrease and reply the first derivative, Kath. So in this case, we will apply the product rule. This will give us ah two x our necks plus the driver of a land of X is one over X and then sits just X squared over, accepted just x and then we can factor out a X This will give us, um I'LL give it to Alan. Act close one. And then we set this whole thing equal to zero. We get access, he cool too. Zoo And and then we have to find out where twelve and experts one is equal audio too. Plus one way. Get on. Actually cool with negative one half. Go on, then access we take way, apply exponential function need apply and and then we get accidental to eat the negative one half or simply one over disclosure of the So now we're going to apply a sign chart and and essentially we're looking at the values of zero and one of the square root of e. Yeah, way. Ignore values less in jail because their domain is only expert and zero. And when you plug in values between Joe and squared of one of one of the square root of e u Get a negative time, and if you try anything very positive, you get positive. So now that we know the signs of that crime, we know where it is increasing and decreasing. So it is increasing from one over square root of E to infinity, and it is decreasing between zero and one over square. Did he? And we know that it has a local men because it's going from it is decreasing and then increasing. So we have a local monitoring at I won over the square of E X is equal to, and there's no local Max. And now we're going to find the clown cabinet pinpoint, which requires taking the second derivative. So you will apply the change. Will, um you're kind of product. Cool again. But I've already taken to do over there before. And it would simplify down to two our x plus three way said this equals zero. Then we get our necks is equal to negative three, huh? And then we apply our exponent, Tal exponent Tal underside. And this gives us access record too, eh? To the negative you have. So now we all of we will create a sign sure to evaluate where are you function and where our fun cavity is positive or negative. So we're looking at from zero and e to the negative three Have we're going to sign for a half double crime in this case and so few and we ignore these any number lessons there because I don't mean it's great in job. So numbers if you plug in value between Joe and easier negatory, African negative. And if you apply in numbers greater than easy two negatives, we have to get positive number. Since this negative, that is Khan Cave down and this is Kong gave up. So an interval notation um so con cave up, which is are you sign? It's from E to the negative. We have two positive infinity and then conquered Down occurs which is a ah, a upside down. You occurs between zero and E to the negative three half and our inflection point inflexion So inflection point occurs when our sign for con cavity changes with thine which occurs that equals negative. Three. Africa's goes from negative to positive go from e to the minus we have it is sorry x equal in surmised we have and there's no other inflection point because there's no other place where the sign changes, and that is the answer to our question.