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# extremum

Extrema are minima and maxima. The singular forms of these words are extremum, minimum, and maximum.

Extrema may be “global” or “local”. A global minimum of a function $f$ is the lowest value that $f$ ever achieves. If you imagine the function as a surface, then a global minimum is the lowest point on that surface. Formally, it is said that $f\colon U\to V$ has a *global minimum* at $x$ if $\forall u\in U,f(x)\leq f(u)$.

A local minimum of a function $f$ is a point $x$ which has less value than all points “next to” it. If you imagine the function as a surface, then a local minimum is the bottom of a “valley” or “bowl” in the surface somewhere. Formally, it is said that $f\colon U\to V$ has a *local minimum* at $x$ if $\exists$ a neighborhood $N$ of $x$ such that $\forall y\in N$, $f(x)\leq f(y)$.

If you flip the $\leq$ signs above to $\geq$, you get the definitions of global and local maxima.

A “strict local minima” or “strict local maxima” means that nearby points are strictly less than or strictly greater than the critical point, rather than $\leq$ or $\geq$. For instance, a strict local minima at $x$ has a neighborhood $N$ such that $\forall y\in N$, $(f(x)<f(y)\textrm{ or }y=x)$.

A *saddle point* is a critical point which is not a local extremum.

## Mathematics Subject Classification

26B12*no label found*

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