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Numerade Educator

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Problem 60 Hard Difficulty

(a) Use a graph of $ f $ to estimate the maximum and minimum values. Then find the exact values.
(b) Estimate the value of $ x $ at which $ f $ increases most rapidly. Then find the exact value.

$ f(x) = x^2 e^{-x} $

Answer

a) local $\max =\left(2,4 e^{-2}\right)$
local and $a b s \min (0,0)$
b) $x=2-\sqrt{2}$

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Video Transcript

all right. Eh? So we're being asked to use the graph of F to estimate the maxim men's and find the exact values. No, to in order to find the maxim men, we first have to find the first derivative. And we can just first of all, look at this function could see that there is some sort of ah, local Max and Minnick ranked, um, dysfunction as sure I believe diverges to attic. It goes he never goes past the fact that actually it never touches the exact exactly do that wrong. So that means we have a some sort of possibility of pasta. This is possible of our absolute men and Wei have a local max. So we're looking for two different answers. So when we take f crime so it's virtual stakeout. Prior practices, they Prada. Cruel. So this is going to come out to be to x e to the minus x, um, minus X square. Is it a mine effects? And then we can pull out a factor of X and eating my defects. I was going to give us too many thanks. And with that testicle, we'LL get X equals zero and we get X equals to. So just by looking at this picture, we know that the local men is zero. And we know that the local max is that two. So this is f zero will be zero. And we know that effort too will be for e through the mine in two Now to find the values of which efforts increasing with rap. Police saying essentially were the greatest positive slope, um, and was the greatest about positive canyon lunch slope, and we're being asked to find exact values, so this usually occurs when is usually occurs at the inflection point. This is when the second derivative of F X equals zero first gonna find second residence. So this comes out to be two e minus X minus two x the mine effects and then i'LL be again minus two x e minus X and plus X Square Eve minus X so we can pull out effective in minus X, and I'm going to combine these two terms. It'LL be even attack So it'LL be two minus four X um, plus X squared. And this this polynomial right here we set this we set this entire thing because, you know and in order to find the words to this part Norm, you can't be factored so we have to plant the quadratic formula, which is negative. B plus a minus quivered, B squared, minus four A c over to a Where is the coefficient of X square mean of ex whatever in front of the square function and then be infront of the acts in this case is minus four and see it too. So when you do this, you get to plus or minus two so we plug this into our prime because that prime tells us the slope and we're looking for the most deep slope. However, we could have to skip this process from plugging and we can actually look at this graph we going today, okay, since two is our local max, anything above two has to be decreasing in flow, and anything less than two has to be an increasing club. But just by looking at the grass, you know that two minors room too has to be the max value. So two minutes for two is where the steepest political