1. 1 Extreme Values

2. Why find extreme values of functions?
To find where functions are optimized
Find maximum profit
Find minimum cost
Find best dose of medicine
Find best gas mileage per speed
Etc….

3. Extrema (plural of extreme)
Either minima or maxima
Y values
Occur at either x=c or at the point (c, f(c))
Absolute or global extrema are the highest or lowest y value for the whole function – there can be at most one for each type
Relative or local extrema are a high or low value relative to the values around it – there can be more than one of either type
If no interval is specified in the problem, then we assume we are talking about the whole domain of the function
Relative extrema CANNOT occur at the endpoints of an interval.
Extrema do not have to be nice, tidy hills and valleys, they can be corners or cusps or discontinuities

4. Extreme Value Theorem (EVT)
If f(x) is continuous on a closed interval [a,b], then f(x) has both a max and a min within that interval

5. So where do these extreme values occur?
The theorem doesn’t help us with that!
The graph of f(x) is shown below. What is the value of the derivative where the relative min and max occur?
If the derivative exists at the extreme value, f’ will always be = 0

6. Can relative extrema occur at any other place other than where f’(x)=0?
Example 2: Sketch the graph of
and graphically find any relative extrema.

7. Critical Values
A critical value of a function is a value x=c such that f’(c) = 0 OR f’(c) DNE.
Relative extrema can only occur in an open interval at a critical value.
Absolute extrema can only occur at a critical value OR at an endpoint of an interval.

8. Steps for finding absolute extrema
Find critical points (f’(c)=0 or does not exist)
Evaluate f(x) at each critical point
Evaluate f(x) at each endpoint of interval
The smallest # from steps 2 and 3 is min
The largest # from steps 2 and 3 is max

9. Example 3
Determine if the EVT applies. If so, find the extrema of on the interval [-1,2].

10. Example 4
Determine if the EVT applies. If so, find the extrema of on the interval [-8,1].

11. Example 5
Determine if the EVT applies. If so, find the extrema of on the interval

12. Example 6 – What do we do if there are not endpoints?
Find the extrema of Use your calculator to verify.

13. Example 7
Find the extrema of

14. Example 8
It’s worth pointing out that even though the sufficient conditions of the EVT are met (the “if” part), a function may still have both an absolute max and min.