# 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.