How do you find the inflection point of the function #f(x)=xe^(-2x)#?

1 Answer
Jul 12, 2018

Below

Explanation:

You can find the inflexion point of a function by first finding the second derivative and letting the equation equal to 0.

#f(x)=xe^(-2x)#
#f'(x)=xtimes-2e^(-2x)+e^(-2x)#
#f'(x)=e^(-2x)(1-2x)#
#f''(x)=e^(-2x)times(-2)+(1-2x)(-2e^(-2x))#
#f''(x)=-2e^(-2x)-2e^(-2x)+4xe^(-2x)#
#f''(x)=4xe^(-2x)-4e^(-2x)#

To find the inflexion point, #f''(x)=0#

#4xe^(-2x)-4e^(-2x)=0#
#4e^(-2x)(x-1)=0#
#4e^(-2x)=0# or #x-1=0#

#4e^(-2x)=0# has no solution because #e^(-2x) >0#

#x-1=0#
#x=1#

Test #x=1# (YOU MUST TEST YOUR POINT FOR CONCAVITY)

When #x=0#,
#f''(0)=-4#
When #x=1#,
#f''(1)=0#
When #x=2#,
#f''(2)=0.07326...#

Therefore, there is a change in concavity and so there is a point of inflexion at #(1,e^(-2))#