For what values of x is #f(x)= xe^-x # concave or convex?

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Answer 1 · 2017-02-05T13:28:08

#f(x) = xe^(-x)# is concave in #(-oo,2)# and convex in #(2,+oo)# having an inflection point in #x=2#

We need to solve the inequality:

#f''(x) > 0#

so we start by calculating the second derivative of the function:

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

#f'(x) = e^(-x) -xe^(-x) = e^(-x)(1-x)#

and again:

#f''(x) = -e^(-x)-e^(-x)(1-x) = e^(-x)(x-2)#

Now to solve the inequality we need to consider that:

#e^(-x) > 0# for every #x#, so:

#f''(x) >0#

#e^(-x)(x-2) > 0#

#(x-2) > 0#

#x > 2#

Thus #f(x)# is concave in #(-oo,2)# and convex in #(2,+oo)# having an inflection point in #x=2#

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