The special case of x⁴

Key Questions

  • How do you find the stationary points for #f(x)=x^4# ?

    The stationary point is #(0,0)# and it is a minimum point.

    The first task in finding stationary points is to find the critical points, that is, where #f'(x)=0# or #f'(x)# DNE:

    #f'(x)=4x^3# using the power rule
    #4x^3=0#
    #x=0# is the only solution

    There are 2 ways to test for a stationary point, the First Derivative Test and the Second Derivative Test.

    The First Derivative Test checks for a sign change in the first derivative: on the left the derivative is negative and on the right the derivative is positive, so this critical point is a minimum.

    The Second Derivative Test checks for the sign of the second derivative:

    #f''(x)=12x^2#
    #f''(0)=0#

    There is no sign, so the second derivative doesn't tell us anything in this case.

    Since there are no other minimums and #lim_(x->-oo)f(x)=lim_(x->oo)f(x)=oo#. The stationary point is an absolute minimum.

    Amory W. · · Aug 25 2014

Questions

Graphing with the Second Derivative