Review of critical points of functions of one variable We already know the concepts of critical point, maximum and minimum for functions of one variable. The function $f(x)$ has a local minimum in $x=a$ when the graph near $x=a$ is above $f(a)$, more precisely, if there is an interval $I$ around $a$ such that $f(x)\ge f(a)$ for all $x$ in $I$.

A minimum (or maximum) of a differentiable function $f(x)$ is always a critical point, that is, a point $x=a$ at which the tangent line of $f$ is horizontal, in other words $f'(a)=0$. A sufficient condition for a minimum of $f(x)$ at $x=a$ is the second derivative criterion: if $f'(a)=0$ and $f''(a)>0$, then $f(x)$ has a local minimum in $x=a$.

With a second order Taylor approximation of the function $f$ at $x=a$ we can understand this as follows: if $x=a$ is a critical point, then the Taylor polynomial of degree 2 at $x=a$ is equal to

$$f(a)+\tfrac{1}{2}f''(a)(x-a)^2\tiny.$$ So the function is near $x=a$ almost the same as a valley parabola with minimum $x=a$ in case $f''(a)>0$. If $f''(a)<0$, then we are dealing with a mountain parabola with a maximum at $x=a$.

If $f'(a)=0$ and $f''(a)=0$ you cannot conclude what kind of critical point $x=a$ is: it might be a maximum or minimum, but it could also be an inflection point, i.e., a point where the derivative has a maximum or minimum. The figure below illustrates the three special points of a function of one variable.