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:

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

Global (or Absolute) Maximum and Minimum

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

Assuming this function continues downwards to left or right:

• The Global Maximum is about 3.7
• There is no Global Minimum (as the function extends infinitely downwards)

Calculus

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