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.

Local Max and Min

We can see where they are,
but how do we define them?

Local Maximum

First we need to choose an interval:

Local Max 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: f(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.

Global Max

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.