How do you determine whether the function f(x) = xe^-x is concave up or concave down and its intervals?

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To particularize angularity, dissect the mark of ##f''(x)##.

##f(x) = xe^-x##

##f'(x) = (1)e^-x + x[e^-x(-1)]##

## = e^-x-xe^-x##

## = -e^-x(x-1)##

So, ##f''(x) = [-e^-x(-1)] (x-1)+ (-e^-x)(1)##

## = e^-x (x-1)-e^-x##

## = e^-x(x-2)##

Now, ##f''(x) = e^-x(x-2)## is faithful on its estate, ##(-oo, oo)##, so the barely way it can veer mark is by cessation through cipher. (The barely enclosure collection are the ciphers of ##f''(x)##)

##f''(x) = 0## if and barely if either ##e^-x=0## or ##x-2 = 0##

##e## to any (real) command is enacted, so the barely way for ##f''## to be ##0## is for ##x## to be ##2##.

We enclosure the compute line:

##(-oo, 2)## and ##(2,oo)##

On the season ##(-oo,2)##, we possess ##f''(x) < 0## so ##f## is retreating down.

On ##(2,oo)##, we get ##f''(x) >0##, so ##f## is retreating up.

Inflection purpose

The purpose ##(2, f(2)) = (2,2/e^2)## is the barely flexion purpose for the graph of this operation.

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