Question #db8e2

1 Answer
Jul 10, 2016

For increasing you have to find out where the first derivative is positive, for decreasing where first derivative is negative

Explanation:

Let's calculate the first derivative of the function #f(x)#:

#[(x+2) * e^-x]'# = #(x+2)' * e^-x+(x+2)*(e^-x)'=1*e^-x+(x+2) (-e^-x)=e^-x-xe^-x-2e^-x=-xe^-x-e^-x=e^-x (-x-1)#

Now, regardless of the value of #x#, #e^-x# is always positive, so the sign of the first derivative depends on whether #(-x-1)# is positive or negative. That is:

  • If #(-x-1)>0#, the first derivative is positive and the function is increasing. Thus, if #-1>x# the function is increasing
  • Conversely, if #(-x-1)<0#, the first derivative is negative and the function is decreasing. Thus, if #-1 < x# the function is decreasing