How do you determine the interval(s) where the function #f(x) = xe^-x# is concave down?

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Answer 1 · 2015-04-30T07:55:57

A function is concave down in the intervals in which its second derivative is negative.

So:

#y'=1*e^-x+xe^-x*(-1)=e^-x-xe^-x#

#y''=e^-x*(-1)-[1*e^-x+xe^-x*(-1)]=#

#=-e^-x-e^-x+xe^-x=xe^-x-2e^-x=#

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

#y''<0rArrx<2#.

(this is because the exponential function is always positive)

It's clear also seeing the graph:

graph{xe^-x [-10, 10, -5, 5]}

The point #A(2,2e^-2)# is an inflection point.