How do you find the extrema for #f(x)=x^4-18x^2+7#?

1 Answer
Jun 17, 2015

Answer:

This function has 3 extrema:

  • A maximum at 0 #f(0)=7#
  • 2 minima at -3 and 3 #f(-3)=f(3)=-74#

Explanation:

To calculete the extrema of a function you have to find points, where #f'(x)=0# first.
In this case you get:
#4x^3-36x=0#
#4x(x^2-9)=0#
#x=0 vv x=-3 xx x=3#

Now you have to check how #f'(x)# looks like in the surrounding of the points calculated above.
To check the behaviour you can either draw a graph or calculate #f''(x)#

  1. If #f'# changes sign from positive to negative or #f''<0# then it is a maximum
  2. If #f'# changes sign from negative to positive or #f''>0#- it is a minimum
  3. If #f'# does not change sign then there is no extremum at this point.