Fermat's theorem

Fermat’s theorem links relative maxima and minima to the vanishing of the derivative: at interior extremum points, if $f$ is differentiable, the tangent is horizontal.

Theorem — Fermat

If $f$ has a relative maximum or minimum at an interior point $x_0$ of its domain and $f$ is differentiable at $x_0$, then $f'(x_0) = 0$.

Remark — Geometric meaning

At a relative maximum or minimum point the tangent to the graph is horizontal. The points where $f'(x_0)=0$ are called stationary points or critical points. But beware: not all critical points are extrema (example: $f(x)=x^3$ has $f'(0)=0$ but $x=0$ is an inflection point, not an extremum).

At the relative maximum point the tangent is horizontal: $f'(x_0)=0$.

Links

Topics: Derivatives
Concepts: Maximum minimum · Stationary point · Fermat’s theorem
Skills: Study a function
People: Pierre de Fermat