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

• How do you find the stationary points for `f(x)=x^4` ?
• How do you find the inflection points for `f(x)=x^4` ?
• How do you find the relative extrema for `f(x)=x^4` ?
• How do you determine the concavity for `f(x)=x^4` ?