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A function f has a local maximum (or relative maximum) at c if f (c)

≥

f (x) when x is near c [this means that f (c)

≥

f (x) for all x in some open interval containing c]. Similarly, f has a local minimum at c if f (c)

≤

f (x) when x is near c

Example:

1. f (x) = x

2

. Identify min and max values.

2. f (x) = x

3

. Identify min and max values.

Note: If f has a local max or min at c, then c is a critical number of f

The Closed Interval Method: To find the absolute max and min values of a continuous function f on a closed interval [a , b]

1. Find values of f at the critical numbers of f in (a , b)

2. Find the values of f at the endpoints of the interval

3. The largest of the values from steps 1 and 2 is the abs. max values value, the smallest of these is the abs. min value

Rolle's Theorem: Assume that f(x) is continuous on [a , b] and differentiale on (a , b). If f(a) = f(b), then there exists a number c between a and b such that f ' (c) = 0

Example 7: Show that f (x) = x

3

+ 9x -

4 has at most one real root.