How do you use the first and second derivatives to sketch # f(x) = (x+1)e^x#?

1 Answer
Nov 2, 2016

there is a minimum in #(-2;-e^{-2})#
and an inflection point in #(-3;-2e^{-3})#
please see the graph below

Explanation:

#f'(x)=e^x+(x+1)e^x=e^x(x+2)# whose only root is clearly in #x=-2#, besides for #x>-2# it is crescent whereas for #x<-2# it decrescent so that #x=-2# is point of minimum whose ordinate is #y=-e^{-2}#

By deriving once more we get #f''(x)=e^x+(x+2)e^x=e^x(x+3)# whose only root is for #x=-3#. So for #x> -3# the function has a positive concativity whereas for #x<-3# the concativity is negative graph{e^x*(1+x) [-4.302, 0.698, -1.02, 1.48]}