Function Variations and Extrema

Variations of a Function

Definition Increasing and Decreasing Functions

Let $f$ be a function defined on an interval $I$.

$f$ is increasing on $I$ if for all $u$ and $v$ in $I$: $$\text{if } u \leq v, \text{ then } f(u) \leq f(v)$$ An increasing function preserves the order.
$f$ is decreasing on $I$ if for all $u$ and $v$ in $I$: $$\text{if } u \leq v, \text{ then } f(u) \geq f(v)$$ A decreasing function reverses the order.

Definition Variation Table

To summarize the variations of a function, we use a variation table. Arrows are used to indicate whether the function is increasing ($\nearrow$) or decreasing ($\searrow$).

Example

The function $f$ is defined on $[-2, 1]$ by the graph below. Construct its variation table.

Answer

The function increases from $x=-2$ to $x=-1$, then decreases from $x=-1$ to $x=1$.

Extrema: Maximum and Minimum

Definition Maximum and Minimum

Let $f$ be a function defined on an interval $I$, and let $a$ be a number in $I$.

$f(a)$ is the maximum of $f$ on $I$ if for all $x$ in $I$: $f(x) \leq f(a)$.
$f(a)$ is the minimum of $f$ on $I$ if for all $x$ in $I$: $f(x) \geq f(a)$.

The term extremum refers to either a maximum or a minimum.

Example

The function $f$ is defined on $[-2, 1]$ by the graph below.

The maximum of $f$ is $3$. It is reached at $x=-1$.