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