Higher-Order Derivative Test

As we know Second derivative test is inconlusive when for critocal point `c` `f''(c)=0`.

In this case we need to use another test.

Higher-Order Derivative Test. Suppose that `k` (`k>2`) is the smallest number for which `f^((k))(c)!=0`. If `k` is and odd number, then there is no maximum or minimum at `c`. If `k` is even number then `c` is maximum if `f^((k))(c)<0`, and `c` is minimum if `f^((k))(c)>0`.

Example. Find and classify extrema of the function `f(x)=e^x+e^(-x)+2cos(x)`.

`f'(x)=e^(x)-e^(-x)-2sin(x)`.

`f'(x)=0` only when `x=0`.

So, there is only one stationary point `x=0`.

`f''(x)=e^x+e^(-x)-2cos(x)`. Since `f''(0)=0` second derivative test is inconclusive.

`f'''(x)=e^x-e^(-x)+2sin(x)`. Since `f'''(0)=0` then we can't say anything about point `x=0`.

`f^((4))(x)=e^x+e^(-x)+2cos(x)`. Since `f^((4))(0)=4>0` and order of derivative is even number then `x=0` is minimum according to Higher-Order Derivative Test.

Note, that there are still examples of non-constant functions whose derivatives of all orders at critical point equal 0.