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# Stationary points

A stationary point $\xo$ of a function $f$ has horizontal tangent $$f'(\xo) = 0$$

A point of inflection is a stationary point that is not an extremum.

In practice, a stationary point $\xo$ is

• a local maximum if $f''(\xo)\lt 0$,
• a local minimum if $f''(\xo) \gt 0$,
• a point of inflection if $f''$ changes sign around $\xo$.

$0$ is a stationary point of $$f_1(x) = x^2,\quad f_2(x)= x^3,\quad f_3(x) = -x^2.$$ It is minimum of $f_1$ ($f''_1(0)= 2$), a maximum of $f_3$ ($f''_3(0)= -2$) and an inflection point of $f_2$ ($f''_2(x) = 6x$).

We show that, when $f''(\xo)\lt 0$, $\xo$ is a local maximum. $f'$ is decreasing and $0$ at $\xo$. On the left of $\xo$, $f'$ is positive, so $f$ is increasing, so $f(x)\le f(\xo)$. Similarly, on its right, $f(x)\le f(\xo)$.

Local minimum (A), inflection point (B), local maximum (C) of a function