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

To determine concavity, analyze the sign 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 continuous on its domain, ##(-oo, oo)##, so the only way it can change sign is by passing through zero. (The only partition numbers are the zeros of ##f''(x)##)

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

##e## to any (real) power is positive, so the only way for ##f''## to be ##0## is for ##x## to be ##2##.

We partition the number line:

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

On the interval ##(-oo,2)##, we have ##f''(x) < 0## so ##f## is concave down.

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

Inflection point

The point ##(2, f(2)) = (2,2/e^2)## is the only inflection point for the graph of this function.