EXTREMA ON AN INTERVAL Section 3.1

When you are done with your homework, you should be able to… Understand the definition of extrema of a function on an interval Understand the definition of relative extrema of a function on an open interval Find extrema on a closed interval

EXTREMA OF A FUNCTION

DEFINITION OF EXTREMA Let f be defined on an open interval I containing c. is the minimum of f on I if for all x in I. 2. is the maximum of f on I if

EXTREMA CONTINUED… The minimum and maximum of a function on an interval are the extreme values, or extrema of the function on the interval The singular form of extrema is extremum The minimum and maximum of a function on an interval are also called the absolute minimum and absolute maximum on the interval

EXTREMA CONTINUED… A function does not need to have a maximum or minimum (see graph) Extrema that occur at endpoints of an interval are called endpoint extrema

THE EXTREME VALUE THEOREM If f is continuous on a closed interval then f has both a minimum and a maximum on the interval.

DEFINITION OF RELATIVE EXTREMA If there is an open interval containing c on which is a maximum, then is called a relative maximum of f, or you can say that f has a relative maximum at If there is an open interval containing c on which f is a minimum, then is called a relative minimum of f, or you can say that f has a relative minimum at

Find the value of the derivative (if it exists) at the indicated extremum. 0.0

Find the value of the derivative (if it exists) at the indicated extremum. 0.0

Find the value of the derivative (if it exists) at the indicated extremum.

DEFINITION OF A CRITICAL NUMBER Let f be defined at c. If , then c is a critical number of f. If f is not differentiable at c, then c is a critical number of f.

Locate the critical numbers of the function. None of these

Locate the critical numbers of the function. None of these

THEOREM: RELATIVE EXTREMA OCCUR ONLY AT CRITICAL NUMBERS If f has a relative maximum or minimum at , then c is a critical number of f.

GUIDELINES FOR FINDING EXTREMA ON A CLOSED INTERVAL To find the extrema of a continuous function f on a closed interval , use the following steps. Find the critical numbers of f in c. Evaluate f at each critical number in . Evaluate f at each endpoint of . The least of these outputs is the minimum. The greatest is the maximum.

The maximum of a function that is continuous on a closed interval can occur at two different values in the interval. True False