Fermat's theorem (stationary points)

In calculus, Fermat's theorem states that if a function has a local maximum or minimum at some point, then the derivative at that point must be zero if it exists.

Proof

Suppose $f(x)$ is a function that has a local maximum at $x=x_{0}$.

The definition of a local maximum says there is some range $a<x_{0}<b$ such that $f(x)\leq f(x_{0})$ if $a<x<b$. From this, we know that when $a<x<x_{0}$,

${\frac {f(x_{0})-f(x)}{x_{0}-x}}\geq 0$

and when $x_{0}<x<b$,

${\frac {f(x_{0})-f(x)}{x_{0}-x}}\leq 0$

The definition of the derivative says

${\frac {df}{dx}}=\lim _{x\to x_{0}^{+}}{\frac {f(x_{0})-f(x)}{x_{0}-x}}=\lim _{x\to x_{0}^{-}}{\frac {f(x_{0})-f(x)}{x_{0}-x}}$

By the above, these difference quotients can only be equal if they are both 0. This is Fermat's theorem.

If $f(x)$ has a minimum at $x=x_{0}$, then $-f(x)$ has a maximum at $x_{0}$. Then by the above, ${\frac {d}{dx}}(-f(x))=-{\frac {df}{dx}}=0$ so the derivative of $f(x)$ must also be 0.