4.1 HW Problems

Table of Contents:

• This section shows how to locate and identify extreme values (both minimum and maximum) of a function from its derivative.

Definitions

• Let f be a function with domain D.
• So f has an absolute maximum value on D at a point c if

$$ f(x)\leq f(c) \enspace \,\,\, \text{for all x in D} $$

• And an absolute minimum value on D at c if

$$ f(x)\geq f(c) \enspace\,\,\, \text{for all x in D} $$

• Maximum and minimum values are called extreme values of the function f.
  • absolute maxima or minima also called “global maxima or minima”
• Functions defined by the same equation or formula can have different extrema (maximum or minimum values), depending on the domain. A function might not have a maximum or minimum if the domain is unbounded or fails to contain an endpoint.

Example: The absolute extrema of the following functions on their domains can be seen below. (Notice how the functions have the same defining equation, $y=x^2$ ,but the domains vary.)

(GRAPH 1) See how some of the functions DON’T have a max or min value. We will now see a theorem that says that a function that is continuous over (or on) a finite closed interval [a,b] has an absolute max and and absolute min value on the interval.

Theorem 1—The Extreme Value Theorem

• If f is continuous on a closed interval [a,b], then f has BOTH an absolute max value M and an absolute min value m in [a,b].