For a function of one variable, f(x), we find the local maxima/minima by differenti- ation. Maxima/minima occur when f (x) = 0. x = a is a maximum if f (a) = 0 and f (a) < 0; • x = a is a minimum if f (a) = 0 and f (a) > 0; A point where f (a) = 0 and f (a) = 0 is called a point of inflection.

Locating Local Maxima and Minima (Necessary Conditions) It states: Every function which is continuous in a closed domain possesses a maximum and minimum Value either in the interior or on the boundary of the domain. The proof is by contradiction.

When a function’s slope is zero at x, and the second derivative at x is: less than 0, it is a local maximum. greater than 0, it is a local minimum.

The Closed Interval Method

1. Find all critical numbers of f within the interval [a, b].
2. Plug in each critical number from step 1 into the function f(x).
3. Plug in the endpoints, a and b, into the function f(x).
4. The largest value is the absolute maximum, and the smallest value is the absolute minimum.

An absolute maximum point is a point where the function obtains its greatest possible value. Similarly, an absolute minimum point is a point where the function obtains its least possible value.

On a closed interval, the justification of an absolute maximum or minimum can be accomplished by identifying all critical values as well as the endpoints, evaluating the function at each of these values, and then identifying which value of x corresponds to the absolute maximum or minimum of the function.

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. The Global Maximum is about 3.7.