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]}