1. Unit 4: Applications of Derivatives
Sec 1: Extrema and Critical Points

2. I. Definition of extrema
Also called Extreme Values
Let f be a function defined on an interval, I, containing c.
1. f(c) is the maximum of f on I if f(c) ≥ f(x) for
all x in I.
2. f(c) is the minimum of f on I if f(c) ≤ f(x) for

3. Extreme value theorem
If f is continuous on a closed interval [a, b], then f has both a maximum and a minimum on the interval.
Ex. Graph y = x² + 1 on the intervals [-1, 2] and (-1, 2) to demonstrate.

4. Relative Extrema/Critical numbers
What is the difference between a absolute and a relative maximums and minimums?

5. II. Definition of critical numbers
Let f be defined at c. If f’(c) = 0 or if f is not differentiable at c, then c is a critical number of f.
Critical numbers theorem
If f has a relative max or min at x = c, then c is a critical number of f.

6. Guidelines for finding extrema on a closed interval
Find the critical #’s of f on (a, b) by solving f’(x) = 0 and locating where the Derivative Does Not Exist.
Evaluate f at each critical # in (a, b) by substituting the critical # in f(x).
Evaluate f at each endpoint of [a, b].
Determine which critical value/endpoint is the maximum and minimum.
The least y-value is the minimum
The greatest y-value is the maximum.

7. Ex 1: find the extrema of on the interval [-1, 2].

8. homework
Pg 165 #17-31 odds