1. 3.2: Extrema and the First Derivative Test

2. Definition of Relative Extrema
Let f be a function defined at c.
f(c) is a relative maximum of f if there exists an interval (a, b) containing c such that for all x in (a, b).
f(c) is a relative minimum of f if there exists an interval (a, b) containing c such that for all x in (a, b).

3. Where?
If f has a relative minimum or relative maximum when x = c, then c is a critical number of f.

4. 1st Derivative Test
If c is a critical number of f, then f can be classified as a relative minimum, a relative maximum, or neither:
If changes from negative to positive at x = c, then f(c) is a relative minimum.
If changes from positive to negative at x = c, then f(c) is a relative maximum.
If does not change sign at x = c, then f(c) is not a relative extrema.

5. Find all relative extrema:

6. Find all relative extrema:

7. Find all relative extrema:

8. Absolute Extrema Let f be defined on an interval I containing c.
is an absolute minimum of f on I if for all x in I.
is an absolute maximum of f on I if for all x in I.

9. Extreme Value Theorem
If f is continuous on [a, b], then f has both a minimum value and a maximum value on [a, b].
Absolute max. and absolute min. values can occur at either critical values or endpoints.

10. To find extrema on [a, b] Find critical numbers of f.
Evaluate f at the critical numbers in the interval.
Evaluate f at the endpoints, a and b.
The smallest is the minimum, the largest is the maximum!

11. Example
Find the minimum and maximum values of f(x) = x2 – 8x + 10 on the interval [0, 7].

12. Example
The concentration C (in milligrams per milliliter) of a chemical in the bloodstream t hours after injection into muscle tissue can be modeled by
Determine the time when the concentration has reached a maximum.