The derivative of a function f is given by f'(x)= (x-3)e^x for x>0 and f(x)=7? a.) The function has a critical point at x=3. At this point, does f have a relative minimum or neither?

1 Answer
Dec 9, 2016

Answer:

The function #f(x)# has a local maximum at #x=3#. See explanation.

Explanation:

To find if a function has a critical point at a place where #f'(x)=0# you have to check if the derivative changes sign at this point. If the change occurs then #f(x) has:

  • Minimum if #f'(x)# changes sign from negative to positive
  • Maximum if #f'(x)# changes sign from positive to negative.

To check it you can calculate the second derivative:

#f''(x)=1*e^x-(x-3)e^x=(1-x+3)e^x=(2-x)e^x#

#f''(3)=(2-3)e^3=-e^3<0#

#f''(x)# is negative in #x=3#. This means that #f'(x)# is decreasing at #x=3#, this finally means that #f(x)# has a MAXIMUM at #x=3#