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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^4e^{-x} x $

a) $f$ is increasing on $(0,4) ;$ and is decreasing on $(-\infty, 0)$ and $(4, \infty)$

b) $f$ has local minimum at $x=0$ and $f(0)=0 .$ fhas local maximum at $x=4$ and $f(4)=4^{4} e^{-4}$

c) $f$ is concave up on $(-\infty, 0)$ and $(0,2),$ and $(6, \infty) .$ f is concave down on (2,6). Inflection points are $x=2$ and $x=6$

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Okay, We are being asked to find the intervals in which efforts increasing and decreasing the local Max Amann and its crown cabin inflection point for effort back before we proceed, there is a small error here. This is not so three ex. This's just except for time, nature minus X. All right, guys, for that. So we'LL go right into it. So we're going to find increasing decreasing. We first have to fight first test, which means taking the first river there. So sentences the product. We're going to apply the product well, so this is going to give it for excuse? I'm eating a minus X since you two minus tax, there's a negative x here and we apply changeable differently with negative here and then it remains the same Teach in your back and we can pull out apologize for that And we can pull out a X cube and an E to the negative act so that that is a common factor in this world left leave us with four minus X. Can we set this whole thing equal to zero? And we know that this occurred where X is equal to zero and for so Now we're going to create a sign chart Forgot where this is exactly where this is. Increasing. Decreasing. So this is our number line. When Aparicio put for Look this down, bring us down. Um, I'm gonna include X Q and E to the negative X and for my next fix Oh, together. And we're gonna evaluate the signs. So for the numbers to Westerns, you get negative for excuse positive, positive. And he's a negative, actually, always positive possess an exponential function and then for four minus x b positive, positive and the negative. Now you multiply this to get the sign of a prime. So then we're going to get a negative times a positive sounded positive, negative and positive, positive, negative and then positive times. Negative. Negative. Now that we know the sign in that time, we know what is increasing or decreasing. So it is increasing on between june for between Joe and four, and it is decreasing from negative infinity two, zero and four deposited. Very. And we can also figure out the local mons and acts because we know the directions in which the function is moving, synthesis decreasing and then increasing. We have little men at LegalZoom on Bennett is increasing and decreasing. So then we have a local max X equals four. Now, for a con cavity, we have to take the second derivative cast. So we take the second of every day and that comes out to be x square. I'm e to the minus. X times X square minus X plus twelve. And then you can factor extra. My Zeta plus twelve and I climbed out to be X minus two and times X minus six. He set this whole thing because zero and there is equality. O X equals zero two and six. This is a six, by the way. Not a one. So a six. So then we do the exact same thing. We create a sign chart. We have a the number line and they're bigger and include, um zero, two and six. And since x squared times E zero minus act is always positive, we're not going to include it on then X minus six. No, on Then these two multiplied gives us a sign of f double crime. So all numbers between lessons zero for explains to Gibbs was negative numbers. Buffoons, which will give us negative between two and six is positive and this gives us positive for X minus six systems of negative negative, negative. Positive. Now we multiply so negative. Some negatives, positive negative things. Positive, positive and negative. Yeah, and positive. Positive. Positive. So when asked, double prime is positively No, it is. Khan came up. So Caryn cave up, which is you sign and that occurs from negative infinity two two and from six to infinity and in conclave down, which is up that down you occurred between two and six. So inflection point or where the sign changes for controversies are inflection point and fly occurs at X equal two and six because it's kind of a positive to negative and the negative deposit and that was it.