Local maximum and minimum

Definition

Let $D \subseteq \mathbb{R}^n$, let $f \colon D \to \mathbb{R}$, and let $a \in D$. The function $f$ has a local maximum at $a$ if there is $\varepsilon > 0$ such that $f(x) \le f(a)$ for all $x \in D$ with $\|x - a\| < \varepsilon$. The function $f$ has a local minimum at $a$ if there is $\varepsilon > 0$ such that $f(x) \ge f(a)$ for all $x \in D$ with $\|x - a\| < \varepsilon$. A local extremum of $f$ is a local maximum or a local minimum of $f$. The extremum is strict if the inequality is strict for all such $x \ne a$.

Remarks

A global maximum of $f$ is a point $a$ with $f(x) \le f(a)$ for all $x \in D$. Each global maximum is a local maximum. The function $f(x, y) = x^2 + y^2$ has a strict local minimum at $(0, 0)$, which is also a global minimum. In two or more variables, a local minimum that is the only critical point need not be a global minimum. For $f(x, y) = x^2 + y^2 (1 - x)^3$, the only critical point $(0, 0)$ is a local minimum, but $f(2, 3) = -5 < f(0, 0)$. At an interior local extremum of a differentiable function, the gradient is zero.

Explanations

L1 human

A local maximum is a hilltop: it is the highest point in a small disc around it, but a higher hill can exist far away. A local minimum is the bottom of a valley in the same sense.