Extremum Test

Consider a function $f(x)$ in 1-D. If $f(x)$ has a relative extremum at $x_0$, then either $f'(x_0) = 0$ or $f$ is not Differentiable at $x_0$. Either the first or second Derivative tests may be used to locate relative extrema of the first kind.

A Necessary condition for $f(x)$ to have a Minimum (Maximum) at $x_0$ is

$$f'(x_0)=0,$$

and

$$f''(x_0)\geq 0\qquad(f''(x_0)\leq 0).$$

A Sufficient condition is $f'(x_0) = 0$ and $f''(x_0)>0$ ($f''(x_0)<0$). Let $f'(x_0) = 0$, $f''(x_0)=0$, ..., $f^{(n)}(x_0) = 0$, but $f^{(n+1)}(x_0)\not = 0$. Then $f(x)$ has a Relative Maximum at $x_0$ if $n$ is Odd and $f^{(n+1)}(x_0)<0$, and $f(x)$ has a Relative Minimum at $x_0$ if $n$ is Odd and $f^{(n+1)}(x_0)>0$. There is a Saddle Point at $x_0$ if $n$ is Even.

See also Extremum, First Derivative Test, Relative Maximum, Relative Minimum, Saddle Point (Function), Second Derivative Test