Extrema

A function $f : 𝒟 \to ℝ$ has an absolute (global) maximum at a point $x = ω$ if

\forall x \in 𝒟, f (ω) \geq f (x)

The value $f (ω)$ is an absolute (global) maximum of $f$ .

A function $f : 𝒟 \to ℝ$ has an absolute (global) minimum at a point $x = ν$ if

\forall x \in 𝒟, f (ν) \leq f (x)

The value $f (ν)$ is an absolute (global) minimum of $f$ .

Extreme value theorem

If $f : 𝒟 \to ℝ$ is continuous on a closed and bounded interval $[𝒶, 𝒷]$ then there exists $ω \in [𝒶, 𝒷]$ such that $f (ω)$ is a global maximum and there exists $ν \in [𝒶, 𝒷]$ such that $f (ν)$ is a global minimum.

A function $f : 𝒟 \to ℝ$ has a local (relative) maximum at $x = ω$ if

\exists ℎ \in ℝ^{+} : \forall x \in ℬ_{ω} (ℎ) \cap 𝒟, f (ω) \geq f (x)

The value $f (ω)$ is a local (relative) maximum of $f$ .

A function $f : 𝒟 \to ℝ$ has a local (relative) minimum at $x = ν$ if

\exists ℎ \in ℝ^{+} : \forall x \in ℬ_{ν} (ℎ) \cap 𝒟, f (ν) \leq f (x)

The value $f (ω)$ is a local (relative) minimum of $f$ .

A critical point of a function $f : 𝒟 \to ℝ$ is an interior point $ω \in 𝒟$ such that

either $f^{'} (x) = 0$
or $f^{'} (x)$ is not defined at $ω$ .

If a function $f : 𝒟 \to ℝ$ has a local extremum at an interior point $ω \in 𝒟$ and $f^{'} (ω)$ exists, then $f^{'} (ω) = 0$

Closed interval method

$f (x)$ continuous on $[𝒶, 𝒷]$

find critical points in $(𝒶, 𝒷)$
compute $f (𝒶)$ and $f (𝒷)$
the largest and smallest values are global extrema

Explanations and Examples

try $f_{1} (x) = x^{3}$
try $f_{2} (x) = \frac{1}{x^{2} + 1}$
try $f_{3} (x) = cos x$
try $f_{4} (x) = {\begin{matrix} 2 - x^{2} & x \in [0, 2) \\ x - 3 & x \in [2, 4] \end{matrix}$
try $f_{5} (x) = \frac{x}{2 - x} x \in [0, 2)$

Geometry of local extrema.

\begin{matrix} f_{1} (x) & = & \frac{1}{3} x^{3} - \frac{5}{2} x^{2} + 4 x \\ f_{2} (x) & = & {(x^{2} - 1)}^{3} \\ f_{3} (x) & = & \frac{4 x}{x^{2} + 1} \\ f_{4} (x) & = & x^{\frac{2}{3}} {(6 - x)}^{\frac{1}{3}} \end{matrix}

\begin{matrix} f_{1} (x) & = & x^{2} + 3 x - 2 on [1, 3] \\ f_{2} (x) & = & x^{2} - 3 x^{\frac{2}{3}} on [0, 2] \end{matrix}