What is Maximum, minimum? — Mathematics

Definition 3.12 High School Mathematics · Chapter 3 — Functions

Let $f$ be defined on $D$ and $a \in D$ . The value $f(a)$ is the maximum of $f$ on $D$ when $f(x) \leq f(a)$ for all $x \in D$ ; it is the minimum when $f(x) \geq f(a)$ for all $x \in D$ . On a graph, they are the highest and lowest points of the curve.

The curve of f(x) = (x-1)2 + 2 never goes below the dashed line y = 2, and touches it at x = 1: the minimum of f is 2. — The curve of $f(x) = (x-1)^2 + 2$ never goes below the dashed line $y = 2$ , and touches it at $x = 1$ : the minimum of $f$ is $2$ .

Examples

Example 3.13

Show that $f(x) = (x - 1)^2 + 2$ has minimum $2$ on $\R$ , attained at $x = 1$ . For every $x$ , the square $(x-1)^2$ is $\geq 0$ , so $f(x) = (x-1)^2 + 2 \geq 2$ ; and $f(1) = 0 + 2 = 2$ , so the value $2$ is actually reached. Both facts together prove that $2$ is the minimum.

Example 3.10

Consider the function graphed below on $\intcc{-2}{4}$ .

A function defined on [-2, 4], increasing then decreasing. — A function defined on $\intcc{-2}{4}$ , increasing then decreasing.

Reading the graph: $f$ increases from $f(-2) = -1.5$ up to its maximum $f(1) = 3$ , then decreases down to $f(4) = -1.5$ . Its variation table is

$x$	$-2$		$1$		$4$
$f$	$-1.5$	$\nearrow$	$3$	$\searrow$	$-1.5$

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Examples

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