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

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

Calculus 1 / AB

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 3

Derivatives and the Shapes of Graphs

Derivatives

Differentiation

Applications of the Derivative

Campbell University

Oregon State University

Harvey Mudd College

Lectures

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for this program. Hot eight. We said that if Crimea causes euro so we have two solutions at sea courses here on the exit posted to. And we need to consider the rays up intervals from minus Infinity Theatre's young from 0 to 2 in the front row three impunity over the first into Rome. If prime is negative. So the function is decreasing over the 16th of life Promise brothers here the function is increasing over the last Inderal if promise a negative again. So the function is decreasing. That means we have a local. There is a Noko minimum at X equals zero with with value F zero equals zero and then we have a local make Ciman at X equals two with value of two equals to four times e to the minus two. So those are the local maximum. No commitment in the for part B will want to find um we also find a very off eggs at which have increased most rapidly. Okay, so we look has that if f is increasing own 0 to 2 from our previous result. Now we take the Sikander curative, which is X squared minus for express to comes Italy minus eggs and we set the second of irritating equals zero. So basically, we are looking for the maximum value for a prime over the interval from 0 to 2, that the reason we take the second a dirty piece that the second of purity for back is the first of the purity of off the first of authority for five. So we said the second director, purity vehicles zero um, in in this interview, we only have one solution which is exceed posted to minors root structure so we can see that to Sabine double from 0 to 2 minus root up two in the front. Two miners route up to Teoh too. On the first team football, The 600 curative is, um quoting. So let me is a crime is increasing on the second of all if double promise less than zero. So it's crime is decreasing and we conclude that a crime has, uh, maximal value at X equals 22 miners were it up to

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