Question

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

Answer 1

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.