Calculus
Objectives:
3.1: Extrema on an Interval
Name: ___________________
~ 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.
The minimum and maximum of a function on an interval are the extreme values (extrema) of the
function on the interval. The minimum and maximum are also called the ABSOLUTE minimum and
maximum on the interval. The min. and max. are the highest and lowest y-values of a function.
A function does not have to have a minimum or maximum on an open interval.
THE EXTREME VALUE THEOREM: if a function is continuous (no holes/jumps) on a closed interval
then the function has both a minimum and a maximum
Relative Extrema:
Relative Max. and Min.’s occur on the “hill” or “valley” of a graph.
> if there is an open interval where we have a maximum, then it is called
the relative maximum.
> if there is an open interval where we have a minimum, then it is called
the relative minimum.
Ex.) Find the value of the derivative at the given relative extrema.
f ( x) 
9( x 2  3)
x3
at the point (3, 2)
(First find the derivative)
(then plug in the x-value of the given point/relative extrema)
(if this is truly a relative extrema, the derivative should = 0 or dne)
At relative extrema, the derivative is either 0 or does not exist. The x-values at these points are
called CRITICAL NUMBERS. Critical numbers occur when derivatives = 0 or do not exist.
Relative extrema occur at critical numbers or at endpoints of a closed interval.
(Critical numbers don’t necessarily produce relative extrema! It can be at endpoints too!)
FINDING EXTREMA on a CLOSED INTERVAL
Relative extrema ONLY occur at critical numbers (or endpoints), so to find the extrema on a closed
interval:
1)
Find the derivative
2) Find the critical numbers (find where derivative = 0 or dne)
3) Find the value of each critical number in the original function
4) Find the value of each Endpoint in the original function
5) The least of the values is the Minimum. The greatest value is the Maximum.
Ex) Find the extrema of f(x) = 3x4 – 4x3 on the interval [-1, 2]
(first find the derivative)
(to find the critical numbers, find out where the derivative = 0 or dne)
(evaluate f(x) at each critical number AND at each endpoint)
(the biggest answer is the max, and the smallest answer is the min)