# Maxima and Minima of Functions

## Local Maximum and Minimum

Functions can have "hills and valleys": places where they reach a minimum or maximum value.

It may not be the minimum or maximum for the **whole function**, but **locally** it is.

We can see where they are,

but how do we define them?

## Local Maximum

**First** we need to choose an interval:

Then we can say that a local **maximum** is the point where:

The height of the function at "a" is greater than (or equal to) the height anywhere else in that interval.

Or, more briefly:

f(a) ≥ f(x) for all x in the interval

In other words, there is no height greater than f(a).

Note: a should be **inside **the interval, not at one end or the other.

## Local Minimum

Likewise, a local **minimum** is:

f(a) ≤ f(x) for all x in the interval

The plural of Maximum is **Maxima**

The plural of Minimum is **Minima**

Maxima and Minima are collectively called **Extrema **

## Global (or Absolute) Maximum and Minimum

The maximum or minimum over the **entire function** is called an "Absolute" or "Global" maximum or minimum.

There is only one global maximum (and one global minimum) but there can be more than one local maximum or minimum.

**Assuming** this function continues downwards to left or right:

- The Global Maximum is about 3.7
- The Global Minimum is −Infinity

## Calculus

Calculus can be used to find the exact maximum and minimum using derivatives.